Some people believe that whether you're a man or a woman is based on thoughts in your head, rather than the material biological reality of your sex. They also believe that these thoughts mean you can be neither woman or man, which they call 'non-binary'.
Of course to everyone else this is a rather absurd thing to believe. Like the healing power of crystals or some nonsense like that.
I disagree. Words have meaning, you can't just use your own personal definitions. The modern definition of "gender" is based on the concept of gender identity and includes more than two genders [0][1]. If you want to make your point understood by most people, you should say "there are two biological sexes", although that is also not correct[2].
The dictionary pages you linked to illustrate that there are multiple senses for the word 'gender' in modern use, and that its use as a shorthand for 'gender identity' is not the only one.
Also, as the Wikipedia article you linked discusses, 'intersex' is not a type of sex additional to female and male. It's a word used to group various disorders of sexual development.
[0] https://en.wikipedia.org/wiki/Non-binary_gender