I guess I'm not understanding your confusion here. For most of my life, the most common use of the word "gender" was in the context of grammar, specifically the grammar of instances in foreign languages in which gendered pronouns were applicable to not just persons but inanimate objects as well, for example,
la luna, the moon, or
el libro, the book. The moon is not a male or a female (probably), but people do associate cultural characteristics to objects that tend to color them as masculine or feminine, and these associations have come to produce gendered words and pronouns.
Similarly, for my entire life, driver's licenses have had a small section headed "SEX", and that section is always followed by an"M" or an "F". I've never heard anyone ask "what gender is your newborn?" The birth certificate lists "male" or "female" as a fairly objective description of biological appearances under the same heading.
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So, really, at least for me, both those definitions seem fairly quotidian, if slightly old-fashioned. I mean, there's always room for improvement. But "sex" is biological, and "gender" is cultural, at least in the admittedly fluid vernacular we swim in. No?
Am I misunderstanding something? Perhaps it would be helpful if you could outline for me in what way the definitions ought to be reversed. I'm trying to follow along.
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