The first definition relates to sex.
You do realise that when a word has multiple definitions, it's not just the first one that counts. "Female", per the Merriam Webster definitions, can be used to refer to people with a gender identity the opposite of male.
Most definitions turned up by Google are shorter than you'd expect from a respected dictionary, so there's certainly editing that occurs. There's also the fact that it defines multiple-word terms (like trans woman), which dictionaries tend not to do.
Not really relevant, though. We have multiple dictionaries that are freely accessible, such as the one above which you brought up, and which explicitly refers to gender identity.
In the definition you quoted, "woman" does not solely come after the term "identifies as". It also says, very simply, "a transgender woman". A very simple adjective-subject sentence. Someone who is transgender. And a woman.