A lesbian checking in here.
I’m British, so lesbian culture here might vary from the US.
I don’t know anyone who objects to be called lesbian/a lesbian. I probably use gay woman/lesbian interchangeably. Some lesbians use the word dyke, reclaiming it I suppose. I’m still not keen on it and anyone who isn’t a lesbian certainly shouldn’t use it. Tongue in cheek, we also use ‘sister’, as in ‘Have you met Julie? She’s a sister’. I guess it’s a contraction of ‘scissor sister’, another term for lesbian. A term which should probably be left to lesbians to use.
I think gay men and women experience different kinds of societal problems, impossible to say who ‘has it worse’. I do, however, disagree with the people up thread who say it’s easier for women to show affection towards each other in public - that’s a sure way to attract some fairly nasty comments of the ‘you just need some dick’ kind. There’s a huge difference between what goes on in porn and what two women can do in public.
As for gay men and women disliking each other, I would say there is a small minority who are like this but it isn’t the norm in my social circles at all. When it occurs, I’ve noted it’s often young men, in the first flush of coming out, who have no time for anyone they can’t view as potential sex objects. They usually grow out of it. No doubt there’s some men hating lesbians around, but not any of my acquaintance.
In my experience, gay men and women are drawn together - when you are part of a minority, it’s natural that you cluster together. Our friendship group, including men and women, is very close as very few of us have children, so our friends are our family.