Personally, I hate the term mary sue. It's has no fixed definition and I've seen people literally define it as just "Character I dislike." The real thing it actually boils down to is "I don't believe in this character as presented", which can happen for a number of reasons, so it's always more productive to frame the discussion like that instead.
It means a self-insert fanfiction character AFAIK.
Which is inherently an unprovable accusation, unless you know the author personally. It's a term that can't even be applied to Anakin. What parts of his personality were applied from Lucas? What parts of Edward were taken from Meyer? And even if you somehow know the author, what about well written characters that are also greatly similar to the writer? Are they mary sue's by default as well? What about EVERY other character ever? Any writer will tell you that they put pieces of themselves into their characters, so how is a mary sue any different from a regular character beyond the fact that you don't like them?
It's a shitty, shitty term for a problem that is more complex than any framing of it that I've seen. Call something a mary sue, you have 10 different people thinking 10 different things, often for non-supportable reasons. Say it's a character justification issue and explain why you don't buy it, everyone knows exactly what your problem with the character is.
Here are my previous posts on the mary sue, and the current discussion is exactly what I'm talking about. Half the posts are spent debating what a mary sue even is.
Rather than trying to use an inadequate label, it's better to actually talk about the situations you find unbelievable, and nurture the discussion about whether it really is believable or not.