It seems to me like the argument is premised on an equivocation that neither side seems to really address: one side is referring to sex and the other is referring to gender.

My general definition is “adult female human” which makes sense taking into account how “woman” has been used by almost everyone (until very recently). For example, the “women” in “women’s sports” is referring to the sex of the participants, since sports are sex segregated due to biological differences, not gender differences.

I’d also argue that “man” and “woman” are parallel to the words we use to describe other animals, e.g. buck and doe for deer, cow and bull for cattle, etc. They’re species-specific sex identifying terms.

Those I‘ve talked to on the other side treat “woman” as signifying gender rather than sex, so a person who is trying to be treated as a woman socially should be referred to as such.

I don’t see a problem with generally referring to trans women as “women” in social situations out of politeness; the issue of whether a trans woman is a “real woman” only really comes up when the person’s biology actually matters (e.g. “womens health” and the like).

Edit: typos