Indeed, that one identified sex, not gender, and I would say probably so do nearly all if not all birth certificates, even recent ones.
This may seem strange to modern readers, and saying this doesn't imply that I agree or disagree with this statement, but one upon a time long time ago sex and gender were thought to be basically the same thing.
True, most people seemed to believe so. I'm not sure whether they were correct (though I'm pretty sure they were generally closer to the truth that present-day ideological claims), or whether that was a matter of synonymity or empirical evidence.
When I say 'female' and 'male', I take it that those words are defined ostensively, like most words, and they were so defined in the past. But while usually femaleness and maleness was ascertained by looking at the external sexual organs, gametes seem to be what distinguishes females from males, in humans and other species (to be clear the colloquial meaning is not about gametes, but the sex of an individual (human or not) depends on the gametes they would, under some normal circumstances, produce. It's analogous to how to the meaning of 'water' is not about H2O, but whether a liquid (to make it simpler) is water depends on whether it is composed of H2O, at least primarily).
But let's consider one example, to try to disambiguate this: suppose A is an adult human being, and A has a vagina, a female voice, breasts, etc., and a female mind. However, A has testes (even if underdeveloped), rather than ovaries. I'm inclined to think A is a rare case (there are real cases like that) in which A's sex is male, but A is a woman (so, that's her gender). That would justify chaging a birth certificate - when her condition is discovered - to state that A is male, not female. It's one of those unusual cases in which external sexual organs fail to provide accurate information about gonads.
So, my question is how you would classify A?
Would you say A is female, or male? Woman or man?
Depending on your reply, we might have a case of miscommunication, and then we would have to discuss whether birth certificates that say 'F'
mean that the baby has a vagina, or they mean the baby is female (where 'female' is defined ostensively, and turns out to be tracking gonads), and use the fact that she has a vagina as very strong evidence that she's female. In the vast majority of cases, this will not make a difference, but there is a small number in which it will.