I'll push a bit. The stuff about surgery, identity, transformation is not relevant to a discussion of the inter-relationship between genotype and phenotype. If it is, it's much more complex and distributed across the genome than we can track at the moment. We agree-XX is the typical genotype of a woman. Or we can say the absence of the Y. Or we can say the absence of expression of the SRY region of the Y. So, what is it about the expression of the XX genotype that defines a woman (or the other way around if you got SRY going on)? We had the ability to get pregnant. Fails easily as a definition. Some females are sterile. Post menopausal women are still female. So you go, OK, birth gender, by which I'm guessing you mean some read out of the differentiation of sexual characteristics during development. Which ones-spell it out. Your are just shifting the trait you want to hang your hat on. Try that definition with your wife instead. Might not get you much further. She might say "Oh, so now we have quantifiable elements in our definition". Let's talk about measuring manhood and where you fall on that distribution shall we.
Look, I don't give a rat's ass about the transgender swimmer thing or whatnot. The interesting part is genetics and fairness in sport. Here's a different tact. Take your birth gender XYs and (to use your terminology) take a look at which genes are expressed (in reality, which variants of the genes are inherited and the levels to which they are expressed at different times in different cells). Instead of quantifiable traits of "man" and "woman" take "talent" instead. Humans have very little genetic diversity to begin with. Sequence a million people-it's happening right now. Now, start building genetic correlates that are to some extent associated with the 95th percentile of a trait associated with talent. How might that relate to fairness in sport?
Sure, people are going to go "gaslighting". But that's only because I don't care about what you are aguing about and talking beyond the narrow blinders placed around the conversation.