Can you please explain this a little more, because right now I'm afraid you are simply describing a symptom, not the actual cause. And this half-explanation does not in itself justify the clsim that trans women "are" women, or that sex "is" more than biology on which culture has been laid.
So tell me - because if you truly understand what being trans is, this will surely be easy to answer -
Does having "a cognitive/ psychic difference in the brain, such that your perceive / recognise yourself to be other than your birth sex" mean being one sex but with a deep and aching wish you were the other sex, maybe like a blind person has a deep and aching wish to see, or a lonely little girl has a deep and aching wish that she had been born as one of the popular kids instead?
Or is it actually being, in an innate mental way, in ways we don't yet understand, the other sex, implying that sex is not in fact a descriptor of the body but of the mind?
Because take away the emotional manipulation and neither definition actually justifies the demands being made of women in its name.
Neither definition changes the fact that people with female bodies do exist and do face social and physical consequences because of those bodies, and neither a man's deep feeling that he should have had a female body, nor a man's deep feeling that women don't need to have a female body, changes the embodied experiences and needs and self knowledge of the people who actually have a female body one iota.
Because these are things that are entirely to do with the experiences of women, and so no experience or feeling, no matter how genuine, of a man is relevant to them.
So regardless of which definition you go for, in fact regardless of any definition you go for that places more weight on a man's idea of himself as a woman than the embodied fact of female existence, outside of his own mind, in the reality where all the other women of the world live, his experience of "womanhood" is simply not relevant to who women in the original female sense are and what women in the original female sense need at all.
No definition of woman that is stretched to include male people is more relevant to the needs and experiences and reality of female people than the simple old fashioned sex based definition and there is sinply no way round that.