Ok. Excellent. So you think the people who should be admitted to the women's locker room are those whose behaviors and attributes are based on labels of femininity.
Hmmmm...attributes. I'll just assume that we aren't talking about physical attributes here, because possession of a penis is an attribute, but it isn't associated with femininity. We'll assume it is behavior and mannerisms that matter.
But, your definition goes on:
If someone "identifies with" the gender category of "woman" are they automatically a woman, then? It seems rather unverifiable, unless we assume that the publicly stated gender identify is always identical to the internal perception. It sounds like an argument for self ID as the means of identifying womanhood.
I'm going to leave that for a bit, because I don't want to get too distracted from my questions to Boudicca, although I would gladly accept answers from anyone, not just Boudicca. For that question, we don't want to get hung up on definitions. I'll remind everyone of the question. We all agree on the definitions of cisman, ciswoman, and transwoman, regardless of our definition of men and women.
So, the question was why was it that we would support a ciswoman in her desire to avoid sharing a locker room with cismen, but we should not support her in her desire to avoid sharing a locker room with transwomen?
Ideally, the answer wouldn't depend on the definition of "woman" at all, but if an answer can't be stated without using that word, I suppose your definition above can be used. Respond if you care to.