Gender as a term has existed since the 14th century, the distinction that gender = social/cultural aspects and sex = biological aspects is a recent phenomenon, but still predates the internet. The problem with 'biological male' is it actually doesn't tell you anything, it's just a way of calling someone a man with plausible deniability. Are you talking about chromosomes, sholder-to-hip ratio, hormone levels or any of the other biological stuff we conceptually tie to sex? And what is the BBC referring to when calling this unnamed woman a 'biological male' (they're not referring to anything biological, they are calling this woman a man).

also, right from the article disproving your annoyance…

They're referring to the nurse, not the trans woman. The part I quoted is literally the only time the article refers to the trans woman.