I think people's ignorance are more of a problem , sure you can get rid of all the labels it isn't gonna stop ignorance when it comes to sexuality .
If that is his main issue in the word, that you are labeling yourself 'incorrectly', then he needs to take another look at the world around him.. That is such a trivial thing and why is it his concern what you label yourself as.. If you are a woman and are happy with labeling yourself as gay, then I don't see any problem with that at all. I think there are so many misconceptions about the whole LGBT community and people aren't willing to learn more about it that these people just can't get it into their heads that whatever you label yourself is right, because it is right for you.
I agree, It's not really a gender thing. I am fine with calling myself a lesbian sometimes, but around most straight people; I'd prefer to just say gay. Like it or not, lesbian has a sexual/fetish connotation, and isn't even really seen as an actual sexuality as much as a porno term. Ironically, when I call myself gay around straight men, they actually take me more seriously. It's nobody's business what somebody calls themselves, and you can't argue history over why one term is more correct.
If my mum ever mentions a woman who's homosexual, she always says that they're 'gay', never 'lesbian'. I've always found that interesting. But I don't tend to get too bogged down in the labels, we know what you're talking about, and people should just use the ones they're familiar with, if they want to use labels at all. I don't really label myself much, but sometimes it's just easier than getting into a lengthy discussion about your sexual preferences :/
The same thing happened when my mum told her (40ish year old male friend, and a parent of two young boys) that I was "gay". My mum eventually explained it and he understood and felt embarrassed. It's crazy, huh.
