No agenda here other than scientific accuracy. I'll recommend you read this [peer-reviewed and written by a biologist] paper (Why There Are Exactly Two Sexes), which explains the sex binary:

Across anisogamous species, the existence of two—and only two—sexes has been a settled matter in modern biology

Here I synthesize evolutionary and developmental evidence to demonstrate that sex is binary (i.e., there are only two sexes) in all anisogamous species and that males and females are defined universally by the type of gamete they have the biological function to produce—not by karyotypes, secondary sexual characteristics, or other correlates.

The commenter you're responding to is sadly confused. Nobody (or at least certainly not me) is saying that "a woman is someone that is born with eggs" or that "chromosomes strictly determine what these cells become". They're trying to misinterpret what the scientific consensus is, and I would be wary of their agenda. Reading papers like the one I linked is a much better source than the inaccuracies of the commenter you're responding to. If reading papers isn't your thing, here's another quote from biologists elsewhere in the thread:

In animals and plants, binary sex is universally defined by gamete type, even though sexes vary in how they are developmentally determined and phenotypically identified across taxa.