The phrase I used was "the language of transphobia".
JKR said in her piece that she had read widely on this issue and sought to present herself as a neutral outsider who had read stuff from "both sides".
However the language she used was a red flag that this was not the case.
If you read FB post and the poster repeatedly said "moslem" instead of muslim would that set of alarm bells? Their defence would be that it is a correct alternative traditional spelling. If you suggested they were racist they may say "Islam isn't a race". I have only seen racists insist on using the term "moslem". If they insisted they weren't racist I think oh I got it wrong . I don't think that racists get to self define - someone saying they are not racist is not going to persuade.
Similarly someone saying "I'm not transphobic but..." does not mean I think that person could not possibly be transphobic.
As for the language - I haven't got the blog to hand but JKR's repeated references to trans activists (posting on twitter, especially posting offensive insults is not activism) and her saying "biological" and "natal" women. This is not neutral language. It is the language of "gender critism" aka transphobia. The term Rowling wants is cis. Cisgender is in the OED. The refusal to use a neutral non loaded term is, again, to me indicate that JKR has being read transphobic material.
Comments on "people who menstruate" are dishonest. How often have you seen that or similar terms in the wild? The phrase is used, as I'm sure you all know, to be inclusive. What should it be replaced by? Women? But we know not all women menstruate so discussion of, for example medical research relating to those who menstruate then being relabelled as about women would be wrong and the conclusions would be wrong. Do you think it should be replaced by "women who menstruate"? But that, as I am sure you know would not include trans men and non-binary folk, so again would be misleading and change the meaning re the group the research related to.
Some may think what I've said is absurd.
I think the idea that "terf" is a gendered insult used to silence women is absurd. "Shut up terf" is the equivalent of saying "duck off racist". Many think you shouldnt "debate" racists. Many feel that there is no point debating "gender critical" feminists. Their views are legitimised by being deemed worthy of debate, those views are very entrenched and not open to change and you will end up exhausted, hsving wasted your time and possibly given platform to a trsnsphobe in the process.
Apologies for all the typos/wrong tenses used above - had only just woken up and didn't read it back before posting.