All of those ways include a male and a female as far as I know?

Nope! Sperm has been synthesised effectively from egg and used to fertilise another egg. Successful birth with this method has been achieved in animals, in humans birth has not yet been tried, it's slowed down due to regulations

Gender is what you are describing as socially constructed as per textbook definition.

Your textbooks are clearly outdated. That's fine, sociology and gender theory evolve. Read some queer theory, in particular Judith Butler.

if your battle is just about the terminology of how to classify someone's anatomy

It's not just about terminology, it's about the social constructs that are inexorably linked with that terminology. It's not overcomplication to point out that only a tiny fraction of observable sex (fertility) is actually immutable, and even that won't be for long. All others are both mutable and socially chosen to represent sex.

Out of curiosity do you mind me asking what part of the world you live in?

I do mind, sorry, my points aren't linked to any particular country or culture and so I wouldn't want you categorising them as such, intentionally or not.