First is the problem of circularity. If a woman is defined as anyone who wants to identify as a woman, the term has no objective content, making it circular as it defines itself by itself. Second is the ontological contradiction. It simultaneously says that gender is a social construct therefore fluid, non essential, while still being innate and immutable making it essential. It is logically impossible for a category to be both based on a social construction and a fixed internal essence that can be misaligned with one's body. Third, the ideology destroys the very basis of the word "man" and "woman" because if gender is purely a social construct matter of internal feeling, then the words man and woman become meaningless as they no longer refer to the sexual distinction nature of humans. Please respond to my actual objection instead of making your own case. If you make your own case instead of responding, I will point that out.

Your layered definition actually highlights the contradiction. If sex and gender are distinguished, you are under a logical obligation to provide a positive definition for the gender identity layer that does not rely on the sex layer. If one says they are biologically male but their gender identity is woman, what is a woman? If a woman is defined by one who preforms feminine social roles, that becomes extremely regressive because it reduces the concept of womanhood to something aesthetic or a set of stereotypes in which even modern feminists and liberals disagree with. If woman means one with an internal sense of being woman, we run to the circularity problem again. It is like defining a circle as that which has the internal quality of being circular. The inchorence here is very clear that the word "woman" is being used as a linguistic trick. A placeholder for a feeling that has no objective referent once you remove biology. Another issue of incoherence is that if a male can be a woman by identity then woman can no longer be used as a stable category for medical research, sport fairness, single sex protections ect.

The money argument actually proves me correct. Money is a intersubjective social construct as it depends on a shared objective standard (the treasury or economy). Every one agrees that 100 dollars has value. But gender identity is purely subjective. Money is founded on an collective agreement about external reality. Gender identity is an internal feeling that the person who claims so demands that everyone else accepts this as an objective fact even though it contradicts the objective reality of biology. Furthermore, you argue that meaning survives through context and use. But this doesnt resolve the logical problem. If a word woman can mean adult human female, for a doctor, but mean person in a dress in a dressing room or social setting, and anyone who identifies as a woman in law, then the word is clearly unstable with no clear essence of it. The problem here is that gender ideology treats everything as the same. If woman refers to biological sex, the statement is false, if it refers to an internal sense, it becomes circular, if it refers to a social role, it is a stereotype. It's funny you say that the meaning can blur. But that's really not good when it blurs to the point where the same word have opposite and even conflicting definitions. If a woman can be a person who is entirely male with a male body that makes sperm and has never experienced female puberty, then the word woman is empty and useless. There is no reference.