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.
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.