Interesting... I think I understand where we differ.
For me sex is the objective presentation of how the body will reproduce. In all sexually reproducing organisms there is a sex binary even in plants such as the hollybush that has a female plant (the one with the berries) and the male (the one with the spiky leaves). The female is defined by having a body and reproductive system organised around the larger less motile gametes and the male has the body and reproductive systems organised around the smaller and more motile gametes. I disagree that pyschology and socialisation are any part of sex as plants do not have a nervous system capable of producing emotions.
When you discuss social aspects of sex, this is what I understand as gender. I agree with the theory that you can change the way you present, but if you can present that way then that gender presenation is still part of your sex. For example having a certain asthetic or demonstrating a certain behaviour is a social expression of your sex, rather than being a demonstration that you can change that sex. What I mean by this is a that a male can demonstrate behaviour that is more commonly accepted within female e.g. wearing high heeled shoes, but in doing so he is doing he is not changing sex, or his gender.
I agree that gender can change, I just disagree that it can change outside of the restrictions of the sex. The gender expression of females in 11th century Japan is very different from the gender expression of Canadians in the 21st century, but their sex i.e. the way that their body has grown to reproduce is sitll the same. TI observe that feminism has increased the gender identity of women, and the with decriminisation of male homosexuality and marriage equality it looks like the same widening of acceptable gender expression in men is starting to happen.
I also agree that there the presence of DSD and intersex conditions tells us something about sex. I disagree that it demonstrates that sex is a continum and changeable, but that it offers proof that it is fixed. If you could change your sex these people would not exist into adulthood as they would be successfully treated to be able reproduce etc.
In reality though, all of these points are moot. In humans the biggest difference between males and females is the difference in strength. This fact is not only objectively observable, and subjectively felt, but it is also not able to be changed with any type of transition treatment. This means, that even if a male undergoes treatment meant to feminise them, the research that is coming up demonstrates that they still have a significantly higher level of strength than females. It also true that trans-men undergoing testosterone treatment are unable to develop strength comparable to that of even a weak man.
Therefore my theory is this conflict is not about gender identity but about the diffierence in strength and aggression between males and females. I have watched quite a few different transgender women talk about what it means to them to "live as a woman" from Contrapoints to Blaire White and not one of them has spoken about the feeling that they are not able to physically retaliate if attacked by a man. This is why I reject the idea that sex/gender is pyschological because it oversteps the fundemental daily experience of women navigating a world knowing they cannot fight back against a male over the age of 14.