From what I gather OP, the craic is as follows:
Trans people want the term 'woman' to incorporate biological and trans women. They took offence to JKR statement that people who have periods are women because that insinuates trans women are not women. The term 'cis woman' has been brought about by the trans community to describe a biological woman who identifies as her biological sex.
My own opinion - This is all getting a bit ridiculous.
JKR didn't say all women menstruate. She only said that those who do menstruate are women, biological women. And this is a fact that cannot be disputed. Yes, not all women menstruate, but anyone who does is a biological woman.
A lot of women (including myself) fail to see why women should have their identity altered to suit men who want to take over female pronouns. I am a biological woman therefore I am a 'woman'; why should I have to say I am a 'cis woman'? A man who has transitioned is a trans woman, although as stated above this is a term the trans community are looking to have removed. The issue with this (cis woman terminology aside) is that sex is binary, and so regardless of surgery, hormones or self identification, anyone who is genetically male cannot be a biological woman, and therefore using the term 'woman' to incorporate biological and trans women simply doesn't work.
As previous posters have said, this is sex appropriation and a disintegration of our sex-based protection laws. Why is it ok to appropriate our sex but not ok to appropriate someone else's race or culture? It is, unfortunately, another example of how women and women's hard fought-for rights are still being viewed as unimportant and disposable.
JKRs statement from yesterday can articulate this far better that I ever can, but it my view that is that women are now in a very dangerous position where if we state biological fact in defence of our sex-based rights we face the very real possibility of being labelled as transphobic or a TERF, and facing serious violence and discrimination as a result.
Ultimately, whilst I agree that trans people should be afforded the same rights and protection as the rest of society, it should not not be at the expense or detriment of women and our sex-based safety, anti discrimination and protection laws and rights.