Which part of what I posted was nonsense and why? If you want to challenge what is a fairly straight forward assertion, you need to provide reasoning.
Male and female pertain to sex. Man, woman, boy, and girl pertain to gender identity, or perceived gender identity. Scientists say that sex is not binary but bimodal. Gender identity can manifest itself in a number of ways including 'non-binary'.
The Gender Recognition Act provides the mechanism for those people to apply to the state to have their gender identity recognised.
Given this, the terms 'trans women' and cis women are subsets of the gender identity of people as recognised by the state. Those people who feel that they are non-binary or some other which outside of the binary are not accommodated by the Act (that is to say, the state).
AS argues that trans women make cis women a subset of their own sex. This is not so. It would be a reasonable argument to say that trans women and cis women are subsets of women on the basis of gender identity, which I happen to think they are.
And so your argument that you have borrowed from the GCs fails in reasoning for the reason I've given above, but it doesn't rest there, since GCs say that gender identity does not exist. Therefore it leads to the conclusion that 'women' is a class of people that includes all people with the gender identity of being a woman, and excludes all those who do not have the gender identity of a woman. Therefore Aurora Saab is not a technically a woman; but to be clear she is female and free to identify however she wishes, including being a 'woman'.
Addendum ... see posts by @Newhouse for those ill-conceived remarks of yours about skin colour.