The reason why it is so hard to define what character traits define a Mary Sue is because Mary Sues aren't characters at all - they're stories. This is why adding character flaws to characters doesn't make them not be Mary Sues. This is why flawed characters can be Mary Sues. This is why which character appears to be a Mary Sue can rotate from character to character in a story, as each focus character ends up "being one" in turn.
A Mary Sue can have any set of character traits you want them to have. The real problem is the story they're put in. It doesn't matter how simple or how complex a character you make the protagonist, if the story is still a Mary Sue story, then the story will remain a Mary Sue story. You can put the most interesting, deep character you can imagine into a Mary Sue story, and because the world as well as the story revolves around the protagonist, they "become" a Mary Sue.
Worrying about a character being a Mary Sue is backwards - the problem is never that a character is a Mary Sue, it is that the story is a Mary Sue story. No amount of tinkering with a character will correct the flaw. Even having another character take the lead in some crucial scenes won't fix the problem - it is the plot which is ruining the story, not the character.
It is easy to get lost in the weeds, and think about a character as affecting a story in a certain way; however, it is important to remember that it is the author who is ultimately making everything happen. It is the story itself which is creating the intrinsic Mary Sueness of the plot. While a character in a Mary Sue story may be poorly written, the real suspicion needs to fall on the story itself.
