No, it isn’t. Rowling has specifically taken the train to TERF-town, where it’s no longer just “how do you define this”, but rather scaremongering about trans people, conspiracy theories, and excluding/blocking anyone who doesn’t agree with her completely. Take Steven King for example (the world-famous author). He long supported Rowling, promoted the HP books, and has, AFAIK, literally never said a single word against her. Likewise, Rowling has often been thankful to him, praised him for being a true ally to feminism and so on. Recently, Steven King said “Trans-women are women.”. That’s it. Nothing against Rowling, nothing more extreme than a simple assertion of fairly commonly-held opinion. So she immediately unfollowed him (possibly blocked him, only he’d know), and deleted every post she’d made thanking or praising him (certainly in recent months).
She’s not just arguing definitions, she’s aggressively into pretty far-out TERF stuff, and willing to cut herself off from people to prove her devotion. She’s basically joined a cult. Just arguing definitions isn’t joining a cult, but when you’re cutting people off, and promoting conspiracy theories (or at least “Liking” them on Twitter, and following people who promote them), and scaremongering, you are in cult-land.
It’s also worth noting that definition you’re calling “classic” is scientifically wrong, and has been known to be for most of a century. It’s not accurate, and when this is pointed out, the people who have been insisting it is usually get very upset and start just saying “Well there aren’t many people who are intersex!” (because that makes it okay to shit on them and hurt them, I guess?), but “not many” is between about 1 in 1000 and 1 in 60 depending on the precise definition. And that’s just people with measurable anatomical differences, not ones that impact brain chemistry or the like.
Also no-one has “lost their job” over this - people have lost jobs over deciding to be offensive to customers/clients/co-workers and violate the policies of their workplace. That’s how the world works.