On the definition point - a definition is self referential if it uses the same term in the very definition itself. They are not informative. They don't help us pick out what the concept actually is. Many philosophers would deny that they count as definitions at all.
Take the 'chair' definition. It is true that it is hard to create definitions that include all the necessary and sufficient properties of a things but that isn't needed for a definition to be useful or informative. So the definition of chair tells me that it's a seat, usually with four legs and a back rest which someone can sit in. It's not a perfect definition, but it is sufficiently informative enough that I can exclude certain things from the category for sure (a basket doesn't have a back rest, and isn't for sitting in, so it's not a chair) and can include some things in it. It is useful because it tells me what a chair is.
That's all I ask when I ask what a 'woman' is. How would you describe what a woman is, what are the properties of 'woman' such that I could know whether I am part of that category? I'm not asking for a perfect necessary and sufficient definition (although note that it's really funny - many philosophy textbooks say that women= adult human female is as close to a perfect definition and analytic truth as you can get. they would be surprised to hear it's now controversial.) When you say a woman is anyone who identifies as a woman, I at least know that my search has to begin with people who identify as such. But once I have gathered all those people together, I can still ask - by virtue of what are they identifying as women? What ties them together in some way?
For example, here is an informative definition You might say ' a woman is any one who is, or believes they are, an adult human female.' That would mean that it's a state of being and also a mental state. And then we could get into a real discussion - we could say , well why include people who believe they are adult human females and not just the adult human females? Typically believing you are a member of the category doesn't make you a member of that category. Etc. But until we know what the concept is, we can't meaningfully have any discussion about it.
As for people saying why do we need a definition of woman anyway...I'm reminded of Contrapoints latest video where they say that defining woman is 'mere metaphysics' and no one has time to waste on metaphysics. But metaphysics is just the study of what exists, and the nature of things. Contrapoints videos are all about unpacking the nature of things, so it seems they do a lot of metaphysics!But this reply is also misguided and arrogant. Women are having the very definition of their oppressed class redefined and pulled out from under them. Don't we at least get a say in this? And isn't it useful to know who the women are in order to direct help and resources to them?