I should know better than to wade into this quagmire.
I have seen on more than one occasion the claim that the term Mary Sue has been over used to the point of complete and total meaninglessness. Bella Swan is a Mary Sue because a vampire falls in love with her despite her complete lack of personality, Superman is a Gary Stu because he's pretty much invincible and exist to be a perfect savior, Harry James Potter-Evans-Verres is a Sue because he's smarter than an child has any right to be. Yada, yada, yada.
Well I'm going to argue that Mary Sue is a valid category and further more I will argue that no character, no matter how powerful, pretty, smart, cool, or adjectiriffic* is ever a Sue because of those qualities. Rather, what makes a Sue is the way the setting and the writing relate to them. The core of the Sue is that the world revolves around them. If characters exist they exist only in relation to Sue. If an attractive person of the opposite sex exists he will fall for Sue and his love for her will consume his every thought. Bad people will pick on the Sue, becoming obsessed with one-uping/demeaning/humiliating her. Most character's moral alignments can be judged on how friendly they are to Sue and how quick they are to agree with her opinions. Antagonists will either be unpleasant in some way that is obvious to her or attractive and redeemable. Likewise, all scenes are about Sue whether she is present or not. If she's there then it's all about her. If she's absent the conversations will be about her. If the conversations aren't about her then they're fore shadowing something in her future. No world building, no unrelated character development, it's all Sue all of the time.
Not every narrative that fits these points will host a Sue. Some stories are going to stay in the protagonist's point of view throughout, some are going to have heroes that are really good and wise and disagreeing with them is usually bad. What makes a true Mary Sue is that the story is for the character rather than the character being for the story. Not only is any tension removed (I would argue stories are typically formulaic enough and tragedies rare enough that this is true for a lot of tales anyway) but any notion of coherent causality is stretched if not broken. Sues are about wish fulfillment at our core we all want a life that is interesting and meaningful and entirely more convenient than reality.
9 FAST 9 FERROUS