OK first, words can have different meaning depending on the way they are being used. I was obviously using the word gender to differentiate from biological sex.
And I remind you that it could be a simple synonym.
Hermophroditism or intersex (the more modern term) is actually not that uncommon, its about the same percentage of the population that has red hair, so about 1-2% of people are intersex.
Contested numbers, and with a strict definition, you could lower this to ~0,02% (1/5000). If it's about 0,02%. More serious numbers with not an as reductive definition as the previous would be ~0,4%, or 1/250 person.
And, just because someone has a dominant trait they don't have to pick a binary gender, even though many do.
If it's a synonym of sex, then taking the dominant phenotype seems just legit. If it's about gender dysphoria now, then they could still pick either the other one, or play head or tail or change whenever they want just like I did.
As for the definition of gender: either of the two sexes (male and female), especially when considered with reference to social and cultural differences rather than biological ones. The term is also used more broadly to denote a range of identities that do not correspond to established ideas of male and female.
There you go a different definition. The point is, there are a bunch of different dictionaries with a bunch of different definitions. Words are defined by how people use them, and in the current day and age people are using gender to refer to gender identity and sex to refer to biological sex. So if you needed a definition for this then there you go.
I took the first link I had, I'm not here to do cherry picking or anything else, but it seems that google shows the definition you copy pasted.
Still, I'd argue that this is an anglosaxon/american concept mostly. Futhermore, for the social sciences, you'd have to acknowledge the pertinence of the term, but considering how full of flaws gender studies is (crappy methodology, ideology being blatant...), and how it sometimes attempt to make a clear distinction between the social and the biological despite of the evidences that show that being transgender, for instance, is biologically influenced, I'd be cautious with all of this.
Finally, why do you care if a feature like this is added?
I don't care. I told you that the problem you point out can be seen as a non problem.
It doesn't affect you negatively in anyway, but it would affect other people positively.
Only if you make them care about it and a so called expectation of being male or female. I personally don't give a shit if people think I've "feminine traits/behaviours" or if I can't find the Breton territory among the countries while one could find Palestine, western sahara, French guyana, Réunion... Same for not being able to put it in the language list.
Once again, if you care about only can it be positive, and yet, I honestly find it sad if some group of people need a random website from the internet providing them "psotivie" feelings with a features because of how superficial it is.
Personally, I have and continue to struggle with my gender identity, and it is horrible, it is crushing to have these two options that you feel so disconnected from, you have no answer to how you feel cause there seemlingly isn't one.
I suppose "horrible" is an exagerration, else, I'd suggest you to seek for help because a distress for a feature on the internet seems to be a bit of an extreme reaction to a small event. On my part, I feel very disconnected to France and other countries, yet I don't find it "horrible" not to be able to show off a Kroaz du or a Gwenn ha du here for the simple fact that I know who I am, and that not being able to show it won't change anything to that very fact. Same for my gender obviously since I don'(t care about being labeled as a female.
I am not a boy, but I hate my name and voice and feminity of them
Well, then use your second name? If you have multiple names including male and females, that's just a bonus. Else, get a new one; among which you could eventually pick an epicene name for example (Camille works fine in French). As for the voice, you're the one describing it as feminine and it's up to you to accept yourself the way you are or not, and to eventually change it when it's possible, but I don't see how an option on a small website would change anything to this.
Even, on this website having, having essentially a female persona, it seems small and like it wouldn't affect anything but its invalidating and it just adds to all the confusion
Last but not least, I'm not against having a third option or being able to just not fill that part, but if there is no such feature, then I don't think it's a big deal at all because it could just be seen with another angle.