... The entire notion of gender being determined by self-identification amounts to defining "woman" as "person who regards ta's self as a woman". Note the presence of the word "woman" in the definition of "woman". That's a circular definition. Circular definitions utterly fail to explain what words mean. ...
If I ask you to describe the colors red and blue to a blind person, and you are unable to systematically explain those colors to a blind person, does this mean anything? Does this make the colors red and blue non-existent? Does it mean you are a failure at explaining things?
Haven't ever tried, but I don't think I'd be any more unable to systematically explain those colors to a blind person than I'd be unable to explain bat echolocation to a human who can't echolocate. I'd have to explain about photons and lenses and rods and cones and brain wiring specialized for responding to neuron activation ratios and so forth, and it would provide intellectual rather than gut-level understanding, but it would tell the blind person a whole lot more than saying "Red is things that look red."
What we have are a very small portion of the population that say they feel as if they are the opposite gender. That their identity is that of the opposite gender. It isn't a phase. It isn't a wish. It is engrained in their psyche.
It certainly appears to be engrained in their psyche and not a phase; but why do you say it's not a wish? It sure sounds like they wish they were the other sex. In any event, people are all individuals and claiming it isn't a wish is at the very least an overgeneralization.
So instead of asking folks to explain what they mean when they say a "transwoman is a woman", perhaps you need to step back and ask yourself, what is within the entirety of a gender?
Why would I ask myself? It's not up to me -- it's up to the perceptions of the whole community. A gender is an intersubjective category people divide one another into because as children they observed and learned to imitate the categories their elders divided one another into. It's a self-reproducing meme.
How much is our gender is the chromosomes and how much in the DNA and how much in the neurology?
None, none, and none. English* speakers divide people into genders on the basis of anatomical criteria. We learned these criteria by trial and error, comparing possible criteria to our observations of the examples of people the previous generation categorized as male and female. XY people with androgen insensitivity, as well as women who thought of themselves as men, have been among the canonical examples English speakers have observed earlier generations calling "women", and copied; consequently, chromosomes and neurology were not among the criteria we all learned.
(* Sapir-Whorf is alive and well; how people think is influenced by the structure of our languages. So I make no claims about how equivalent English's "man" and "woman" concepts are to their conventional translations in other languages. I suspect speech communities that don't gender themselves anatomically like English speakers are rather rare, but there are thousands of language communities and I have expertise on only one. (Of course many languages have nongendered pronouns like "Estonian's "ta", but that commonly just means pronouns aren't the place in the language where their gender categories show up.))
And which part of that matters the most in our personal identity?
The neurology matters the most in our personal identity; but to draw any conclusion about gender from that is begging the question. No matter how constantly gender ideologues make believe that gender identity is the same thing as gender and no matter how stridently they bully the rest of us to do likewise, those remain two different concepts and treating them as interchangeable is a vanilla equivocation fallacy.
(Moreover, claims that transwomen have "female neurology" and transmen have "male neurology" invariably turn out to be based on some measurement of some sexually dimorphic brain feature -- but there are
hundreds of sexually dimorphic brain features. To infer that an anatomical male has a female brain from the fact that it falls into the typical female range along one axis, even though it falls into the typical male range on some other axis and hasn't even been measured along a hundred other sexually dimorphic axes, is an exercise in simply
caring more about the one axis than the hundred others. Science does not tell us what to
care about. Consequently, all claims I've seen that neurology makes somebody's brain a sex other than his or her anatomical sex are unscientific claims.)