You still haven’t said what gender is, in the sense of an inherent, permanent property that allows one to coherently say “I was born in the wrong body.” We all know what gender roles are.
You’d be free to argue that this isn’t an inherent property, I suppose, given initial probable-fluidity. But since it seems to settle into being a largely fixed part of your psyche before the point you’re likely to have permanent memories, I’m inclined to view that as a meaningless difference.
It’s at about the same level as arguing whether sexuality is an inherent property, I think? It’s another of those “it might be hardcoded or it might be early-development cultural, but it’s basically impossible to change it so…” things.
Edit: I think fundamentally, if you don't object to treating trans people decently like using their preferred pronouns and name, or getting surgery, I don't think it matters too much. I'm not an expert on the definitions by any means, my primary concern is opposing justifications to mistreat trans people, so if it's just a matter of terminology I guess my only suggestion is to read into what the relevant fields of study have to say if it's something you want clarity on.