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.