There are only two genders:male and female. Each person has different personality traits and some of them are more masculine and others more feminine. Nobody is completely masculine or feminine, so each degree of masculinity or femininity does not constitute a new gender. People are either more masculine or more feminine, so their gender is either male or female. Some people have a psychological disorder called gender dysphoria where their gender does not match their sex. These people are transgender people. If gender were completely a social construct, gender would not exist, and therefore transgender people would have no need to exist. Yet, transgender people are very real and their condition is very real, so gender must be real.
There are only two genders and there are only two sexes. I have yet to see anyone even name a third gender or sex.
Male and female are the two sexes. Being a hermaphrodite/intersex (having two sets of genitals) is not regarded as a third sex. If you have two sets of genitals, only one of them ever actually work, and all intersex people are either more male or more female. Science does not regard them as being a third sex, but rather being a combination of the two already existing sexes.
As far as gender goes, there are only two genders (as per the definition) and those are male and female. As already stated in this thread, the definition of gender is "the state of being male or female (typically used with reference to social and cultural differences rather than biological ones)."
Genderfluid is not a gender. I'm not saying that being genderfluid doesn't exist, but it is NOT a gender. It is a description of a gender. Genderfluid means that you switch between being both male and female. So a genderfluid is always still either male or female. Genderfluid is just a description of their gender and not their actual gender. Same goes for agender (lack of a gender) and bigender (having two genders). These things are genders by themselves, but rather they are descriptions of genders.
This is the best I've ever seen it put, you've managed to present the two genders argument without sounding like an asshole! I think of it as a faucet, turning on hot and cold makes warm, but the two knobs are separate. Same idea with gender, if you're a mixture of male and female (psychologically or physically) that doesn't mean there's more than 2 genders.