I feel like you are skirting around the edges of this because you don't want the answer to be what the answer so obviously is, but you will get there in the end.
"Women" is a category word, which is used to describe a group of people consisting of roughly half the human population. "Men" is a category word, used to describe the other half.
The purpose of a category or a group is to unite all the people or things which belong together based on certain characteristics they share, and exclude all the people or things which do not share those characteristics.
"Women" has to mean the group of people who are united by the fact that they share the following three characteristics: adult, human and female.
There is nothing else it could possibly mean.
There are no characteristics that women share with trans women which we do not also share with men.
Being female is what distinguishes us specifically from men.
If you remove female from the definition, what is the difference between women and men?
It certainly has nothing to do with identity, because there is no identity that all women share.
And yes, it does need to be defined, because our access to certain spaces and services, as well as our need for certain rights, is based on it.
And no, it can't mean different things to different people, because in order to have rights attached to it, it needs to actually have an objective meaning which is understood in the same way by all people. It makes no sense to say your right to something is conditional upon being a woman, if you're then going to define a woman as anyone who says they are one. It is necessary for there to be some objective criteria, otherwise the word doesn't mean anything.
Nobody who can explain what they believe a woman is thinks a trans woman is one, and nobody who says trans women are women can explain what they believe a woman is.
It has to mean adult human female. There is nothing else it can possibly mean.
Many trans activists have tried to provide an alternative definition, and all of them have failed.