When an author or writer puts themselves into a story they have written as a character.
It is not a character in which the audience inserts themselves or can relate to. It has absolutely no relation to the audience or readers. It is specifically referencing the creator's personal
avatar.
Most stories written in second person are not self inserts.
Some creators will hide the fact that they use these characters by changing their names and a few details. Self Insertion characters are frequently not
even the main character in a story. Author Steven
King has done this in his books on occasion.
Video
game avatars that the player creates could be considered a
type of self insert character if the player tries to create an approximation of themselves for the
game. This is borderline as the player is limited by the creation tools and
game's storyline.
Another borderline example would be creating a Tabletop RPG character. Players are often free to create backstories and design their characters as they see fit, often mirroring themselves as they imagine they would be if they existed in such a world. Though again, the player is limited by
game mechanics, the whims of the
DM, and the
roll of the die.