I talked about author insertion, though in Van's case, we can exclude it. She said in numerous interviews that Vanyel was intended to be an anti-homophobia character. See below.

"While mary sues and gary lous are often characterized as author inserts gone wrong, most published mary sues that I have run across, including Vanyel are not. Lackey actually did make an author insert later on in the Valdemar books, a Herald named Misty, who isn’t really a classic mary sue."

Misty is an extremely blatant and admitted to author insertion.

I think that while many fanfiction mary sues are author insertions, usually written by young teens, I think that most published mary sues usually aren't author insertions at all. They also aren't necessarily examples of bad writing, but a deliberate appeal to the preteen young teen audience.