Thinking about Tom's definition.
I asked for a non-circular definition of "woman" that includes cis-women and transwomen, and excludes transmen.
Tom's definition works. I think.
Earlier attempts included variations on "feminine gender roles", and my objection to those were that they were either circular, because "femnine" gender roles were the ones expected for women or, if not circular, they literally defined women by their behavior, thus saying that women who were insufficiently feminine for society's taste weren't really women at all.
Tom's definition has a different problem. It defines "woman" as an internal state, frequently corresponding to, but not identical with, an external, measureable, reality known as "female". If you expect people to treat you in accordance with your self image, and that image doesn't match reality, then you are delusional. When it comes to women's private spaces, and women's participation on sports teams, what really and truly matters is that external, measureable, reality. Females really are athletically disadvantaged when compared to males. Females really experience sex and pregnancy differently than males. Those are the issues that matter. Tom's definition is either equivalent to "A woman is anyone who thinks she is a woman", which is circular, or it asks us to respect a delusion, or it is of no value. Having defined "woman" according to Tom's definition, we would then take down the signs over the locker rooms that say "men" and "women" and replace them with "male" and "female".