Look up the definition of "Mary Sue" on ten pages, and you'll find eleven answers.
Art by terriblenerd via TvTropes.
Just in case one of the most ubiquitous terms in even casual literaly analysis has eluded you so far, here's a quick recap: by and large, a "Mary Sue" is a character in a story that is perceived to be somehow "overpowered" in a story. The name itself comes from a character actually called Mary Sue from some old Star Trek fanfiction, who was herself unrealistically capable, incredibly beautiful, on good terms with every main character, often with romantic involvements – and all that while still being in her mid-teens. It was satire, of course, but the name stuck.
So if you've been on the internet for a while, you've probably seen your share of inane "fandom drama", where alleged fans of some intellectual property (a TV show, a film, a video game, books, whatever) spiral into endless bickering and argument over some minute detail of said property. One of the bombshells, which almost never fails to elicit a visceral reaction even on the best-regulated forum is the question, "Is X a Mary Sue?"
Inevitably, such a question draws people to begin comparing "power levels" and debate exactly how many important characters like the accused character and how much. No, you see, X can't be a Mary Sue because he was beaten in a duel by Y in season 9, episode 32! You don't get it, the writers let X lose just so Z could comfort him after his defeat, fuelling their unrealistic romance! Their romance is great, what are you talking about? Z is a terrible character, obviously X belongs with K! If Z and K got together, that'd just make X more of a Mary Sue! And so on ad nauseum.
And then there's the types of Mary Sues. Without any need of completeness, let's look at a couple.
A particular class of Mary Sue is called the "Sympathy Sue", and is usually defined as a Sue that is not (necessarily) beloved by the rest of the cast, and may in fact be entirely incapable and clumsy. The sole purpose of a Sympathy Sue is to make the audience like them by eliciting sympathy by cheaply tugging at your heartstrings with their incredible misfortune.
You'll notice that while the Sympathy Sue is a "type" of Mary Sue, its definition is pretty much the opposite of what a Mary Sue is often said to be.
Then there's the elusive "Gary Stu", who is sometimes touted as the opposite of a Mary Sue (whatever that means, given that the Sympathy Sue is already that in a lot of regards). Other times, a Gary Stu is simply the "male equivalent" of the Mary Sue, because apparently this was a term that needed gendering. But sometimes not. And sometimes the Gary Stu is also called the Marty Stu. Sometimes one is the male equivalent, while the other is the incapable equivalent. Sometimes they switch.
It's a mess, honestly.
Simply put, I find this sort of bickering and pedantry over what exactly constitues a Mary Sue to be simply not useful. It's missing the point. Allow me to offer my own simple, catch-all, completely arbitrary definition of a Mary Sue:
A Mary Sue is a character in a fictional work that the story bends around, rather than the character bending for the story.
I think it's a simple-enough description, but the meaning may not be immediately clear, so let's look at a few examples.
Everyone knows Harry Potter, and over the years I've seen him be accused of indeed being a Mary Sue himself. He does have many of the classic Sue traits: he is gifted, capable, immediately liked by most everyone around him, he is exceptional and treated so based on no other justification than that he is, in fact, Harry Potter.
Yes, Harry is surprisingly capable for being a boy wizard who picked up the magical arts yesterday. This is usually contrasted with the utter ineptitude of Ron, who comes from a comically populous all-wizards family. Yes, Harry is famous throughout the wizarding world, having done essentially nothing to earn that notoriety. Yes, Harry is the Chosen Oneᵀᴹ. Yes, he was left a fortune by his conveniently dead parents, and yes, a slew of the best teachers the wizarding world has to offer are waiting for the moment he turns eleven to get the boy up to speed. If you go with the "checklist-style" definition of Mary Sue, you'll notice he ticks quite a few of those boxes.
And yet I would argue that he is not a Mary Sue.
There's a simple reason for this: all of the above, while true, are not done to prop up Harry's character. The story does not contrive itself to make Harry the greatest wizard who ever lived. Harry's fortune and fame are tools used by the author to put interesting challenges ahead of Harry.
The fact that everyone knows exactly who and where Harry is puts him into the evil Voldemort's crosshairs (nevermind the logistics of crosshairs on magic wands). Coming from a prestiguous family means he has a lot to live up to, expectations of him are high, and he is constantly thrust into situations beyond his level because of this.
Harry isn't harrassed by Malfoy to cheaply tug at the audience's heartstrings for sympathy points, but to reinforce that just because Harry is famous, he is not beyond critique and still has plenty of enemies who have plenty of ways to hurt him.
Everything about Harry that might, on a crusory examination, qualify him for being a Mary Sue actually serves the opposite function: they force the character to adapt and change and grow.
From the same franchise, Hermione Granger sometimes gets accused of being a Sue herself, for being so incredibly smart and capable, often outclassing (indeed, let's be honest, often saving) Harry himself. All the while being a total babe!
...it might be easy to forget with the ubiquity of the Harry Potter movies, but Hermione in the books is actually quite plain. I don't think she is described as ugly by any means, but she's scruffy and weird and somewhat dumpy-looking. As for her wizarding skills, consider that she comes from a muggle (non-wizarding) family. Wouldn't you be, like, way into magic too?
Hermione's magical strength doesn't come innately. By and large in the Harry Potter fandom, the third book is often considered the best one of the series, and it's in the third book that a very important plot point becomes that Hermione is literally using time travel so she can attend multiple magic classes at once. Throughout the book, she is constantly exhausted, close to falling asleep; she is filing herself down to the bone just to cram more magical knowledge into her noggin. This pays off handsomely whenever Hermione saves the day (which is admittedly often) not to mention the whole time travel thing which plays an integral part to the plot.
Again, as with Harry, we see that the "Sue-like" traits of Hermione are not inserted to prop up the character, but instead they are there to enrich both the character and the plot.
Another well known example: Luke Skywalker. In the span of a single film, he rises from being a lowly moisture farmer on a desert planet (whatever that is) to being a renowned revolutionary who single-handedly destroys the galactic superweapon of the evil Empire. Hell, he is guided by actual space magic exclusive only to him to do it! By the time the original Star Wars trilogy ends, Luke surpasses even this, becoming a Jedi knight: the last member of a noble order of space wizards who bring peace and prosperity to the whole galaxy. How much more Suey can you get?!
Once again, we see a similar pattern: Luke's admittedly many strengths and "Suey traits" are not arbitrarily contrived by the story to prop up his character, but rather to put challenges ahead of him.
To gain his magic powers, Luke must seek help of the resident desert hobo / space wizard Obi-Wan, who doesn't just hand out his knowledge; he makes Luke practically beg for it. And true enough, he does let Luke play around with his father's lightsabre for a bit, but he doesn't let him actually use it in action until he's made sure that Luke practised with it.
That's what sets Luke apart from a real Sue: he has to earn what he gets. He wasn't born a jedi knight; he had to religiously train to be worthy of the title. He doesn't become the saviour of the galaxy just by existing; he lost his family, had to flee his home, get on the wrong side of the law, join a rebel resistance, and risk mortal danger to blow up the Death Star. He sweated, he fought, he failed, and then triumphed. The story does not bend for Luke; Luke evolves to follow the story.
If you want a good example of a true Mary Sue, look no further than the infamous Bella Swan of Twilight. Now, it's been a hot minute since I've read the book (and even then I mostly skimmed it), but if I recall correctly, it goes something like this: Bella Swan is a Totally Averageᵀᴹ girl of no special skills or charisma or beauty, who nonetheless accidentally and effortlessly becomes the centre of the book's universe.
Various supernatural hunks fall inexplicably in love with Bella through no effort or action of her own; the vampire Edward likes her smell, and the werewolf Jacob just think she's hot, or something. While not especially capable or exceptional, Bella nonetheless overcomes any and all (usually rather meaningless) challenges set before her, and when she can't overcome some problem, her supernatural lovers are there to assist (which happens a lot).
Even dying is no obstacle to Bella, as she's quickly turned into a supernatural beast herself, which incidentally has practically no effect on her life other than being a bit pale, I guess. Any time something threatens Bella's status as the centre of her universe, the story bends over backwards to desperately contrive a victory for her and her friends.
Bella is a fundamentally passive character. The story happens to her rather than Bella taking action to drive the story. At every turn she beats the odds, she is saved, she is loved, she is protected, she is cherished – all without breaking a sweat. Bella is well and truly a Mary Sue. How you classify her further is entirely up to you (you could probably write an essay on what a boring, terrible, arguably harmful character she is), but in my eyes, she – and characters like her – will always be Prime Sue. (There's a new type of Sue for you to play around with!)
Fundamentally, Sues are lazy and boring characters. That's what makes them something to avoid, not their perceived "power level" or skills or relationships or physical beauty or what have you.
The story bends around the Mary Sue, rather than the character bending for the story.
In an earlier blog, I argued that when writing a story, you should start at the end. In it, I specifically make a point that it's best to change the characters and their backstories around to accommodate the story, rather than the other way around. This ties into that philosophy.
If a character is extremely skilled or gifted at something – perhaps they are a great fighter, an expert computer hacker, a silver-tongued smooth-talker, a great wizard, perhaps all of these and more rolled into one – that, alone, does not make the character a Sue. So long as all of the character's gifts are earned, and more importantly, used for the enrichment of the story, to put greater challenges and obstacles before the character for them to overcome, they are not a Sue.
Harry Potter may start out as a famous wizard child with a lot of money in his bank account, and he becomes a very capable wizard himself through the story, but it's all done so that he can take on the increasingly dire threats the author throws at him. At first, he's barely able to survive an encounter with a wild, mindless troll — by the end, he is taking on an entire Wizard Naziᵀᴹ organisation. His growth is both earned and used.
Hermione spends day and night (and sometimes multiple days per day) studying the magical arts because she wants to be more than the "mudblood" muggle that wizarding society casts her has. She struggles, she fights, she grows, and her skills lead to more interesting story beats rather than become a tool to shut story beats down when the author writes herself into a corner.
Luke Skywalker does become a powerful jedi knight in his own right, but he needs to be to take on the evil Empire of literally galactic proportions.
Yet another example: Doctor Manhattan from Watchmen is virtually omnipotent and omniscient, yet he still faces challenges of his own, namely, what does it mean for one's humanity when one becomes a god? Turns out, that's not an easy question to answer even to someone as powerful and wise as the resident glowing blue guy.
Conversely, if misfortune constantly befalls a character – for example, their parents die tragically, they lose their job, their friends abandon them, their wife files for divorce after cheating on them, and their puppy gets run over by a steamroller, all in the same day – that, alone, does not turn the character into a "Sympathy Sue". So long as the character's misfortune plays a role in the story, to give the character something to overcome, that character is not a Sue. If misfortune happens for no purpose, save only to cheaply tug at the audience's heartstrings by piling on the sadness, then the character is a Sue.
Imagine an underappreciated, underpaid high school chemistry teacher with a harpy of a wife and a disabled son and who is diagnosed with cancer... in the US, no less! Is that a Sympathy Sue, or is it the main character of one of the best TV shows of all time?
A character can be weak or strong, innately capable or not, hell, they can be practically omnipotent if you like, but none of these things, in and of themselves, make the character a Mary Sue. As long as a character faces meaningful challenges, has to make meaningful sacrifices, as long as the plot doesn't contrive itself to benefit or hurt them, they are not a Sue.
So the next time you find your favourite fictional character put on trial before a jury of netizens to ascertain their Sue status, remember this simple rule: the only thing that makes a Sue is their ability to bend the plot around themselves. Likewise, if you ever find yourself contriving your story to justify the protagonist's victory, take a step back, and consider whether you've accidentally invented a Sue.
That's an interesting and imminently useful interpretation of the concept. My own general interpretation was "They have everything without earning it yet act like life is misery for no reason". It never occurred to me that there might be multiple "types" of Mary Sues.
I think I like your overarching version more, if only because it applies a practical slant to the concept.