Hang on, I'll try and give you something to work with.
Room full of people, I pick out all the women. Truth is I don't know what's in their pants or what chromosomes they have, you might say I do because someone's dog always barks at their brothers or some such, but the bare truth is that I don't.
Same room of people, I ask people to tell me whether or not they're a woman. Those declaring themselves women may or may not be the same people I thought were women, and whilst they certainly know better than I do what's in their pants, we're not going to be asking them to prove it any time soon.
So which of them are women? The ones I've identified who also identify themselves as women, they're the easy ones, right? Because as long as we're not checking (and in almost every situation where the word is used, we don't) then that is all we have to go on. Because whilst woman may usually refer to an adult human female, what we actually use it to mean is those we both TAKE to be adult human females, and who also confirm that they ARE adult human females. Regardless of their chromosomes. THAT is what the word means. It is a person who both takes themselves to be and is taken by others to be a woman.
You can be frustrated if you think that's circular, or fluffy, or whatever you like, but you have to acknowledge that "woman" in everyday life is not used to describe those with the confirmed characteristic of being of the female sex. It just isn't. And a word's definition should describe how it is ACTUALLY USED by people, not dictate how it should be. A dictionary won't go into this much detail because that is not the purpose of the dictionary, but the way we actually use the word of course takes precedence.
Now what of those people who either I identify as women but who don't themselves, or indeed the other way round? Given that no proof is being asked for or offered, traditionally the self-declaration of the individual is given more precedence as they at least know what's in their pants. And actually were it a situation where it really mattered what their biology was there would usually be a burden of proof. In everyday situations there is not.
But what if I still disagree that they're a woman? Well, the bottom line is they can be a woman in their eyes and not in mine. Particularly if within society as a whole I am in the minority thinking they're not a woman. Because then to most people they satisfy the criteria of both identifying themselves and being identified by others as a woman. Frankly, as long as it is a situation where no proof is required, a woman is someone who both declares they are one and who society accepts as one, regardless of the assertion of any other individual.
Hope that helps! Got a sporting event to get ready for now but maybe back in the evening :)