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 id

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.