It really depends on the story telling in the game... how well the game is set up and the world is presented. If the presentation itself doesn't break away from immersion and can make me feel like the character presented is a real person in the context of the game (not a pinballing 5-year-old/marksman who people trust with their lives despite having to explain things to them that they themselves, and presumably the main character, grew up with), then... I don't think you would call it self-insertion. I am immersed in the game as the character. I don't feel like a detached and separate person controlling my little puppet, I feel I am that character in the game. As this pertains to the Jimquisition, sure, one may feel that it is weird to think like that, but if a game is well presented, and the main character in that game is a woman, and I can feel believably like a woman instead of a caricature, I could get immersed in that. It doesn't matter if its a woman. I've never been a space marine, a WW2 sniper, a race car driver, a ship captain, star or otherwise, or an Italian plumber that grows to 12 feet tall, spits fireballs, surfs turtle shells, and beats the shit out of giant turtle dragons, and I never will be in real life, but that doesn't mean I can't be them in the context of a game world. I probably will never commit war crimes, experience PTSD, kill someone, see a ghost, or get possessed either. This medium is unique. Books tell you about a world, or a situation, or a person, but you don't become that person and experience it yourself. Movies show you a world, a situation, a person, but you don't become that person yourself. In games, you make the choices. You perform the actions. I become FemShep. But here is the important part. FemShep does not become me. For however long I pick up that controller, I do not become a character who is a man sitting on a couch halfway around the world from his family, the woman I love hundreds of miles away and both of us working our asses off for the two times every month we can see each other. I am a strong female captain commanding a team, one of which I am romantically involved with, and all of which I can actually help, who goes out there, finds evil, and does the hard things I have to do to stop it. And if I (as FemShep) falls for a strong, gung ho type who still goes weak in the knees and stuttery when he first thinks about "us", I would order him to my quarters at 2300 hours, tell him to bring his a-game, and it would be hot. And if presented well, I feel their emotions, their successes, their failures, but here is the important thing... that character is not me. If FemShep were me, not only would they crumble when they have to make the hard choices, but they would probably reassign that guy after saying, "You're nice, but you definitely could not turn me on." Oh yeah, and they would not be Fem. You understand what I'm getting at?
I am in the game world, yes, but the character is not me... I am the character. Provided its a good game. I don't know... from the way you explain it, it still doesn't sound like self insertion, as I don't project myself onto the character so much as step into their shoes.