Wellll, like others have said, twink is a long standing term for very feminine presenting gay men, with varying regional connotations as to whether they’re also very thin and/or petite, and it can be used separate from a “flame”, which is a term for the very flamboyant gay men regardless of their presentation or physical appearance.

There’s a lot of overlap between twinks, flames, otters, and queens (which may or may not mean drag queens, depending on where you are). Some people use them interchangeably, others will use them as separate aspects.

Now, burger, I haven’t heard in a long time among gay folks. The only gay specific usage I am aware of was complimentary, as in “mmm, I wanna get a bite of that burger”. It was usually directed at someone with superior buns, though sometimes someone with a lot of meat. But the last time I heard that, I think it was maybe 1999? 98? And afaik, it wasn’t one of those terms that was used everywhere.

Mind you, most of the above is filtered through media, and/or local gay culture in my area. I’ve never lived anywhere but the south, and only in one state. I’ve visited plenty of the east coast, and since I’ve always felt more comfortable around gay strangers than straight strangers, I have had some exposure to other local cultures. But the majority of what I’ve picked up is regional. You’d be amazed how much difference there is in the gay scene between Atlanta, Charlotte, Nashville, Savannah, etc, just here in the south. You go north and things can be even less familiar.

Back in the day, a lot of the slang and culture traveled with drag queens. The ones that would travel for shows carried all the news and culture with them, like fabulous bards.

Anyway, memory lane aside, that’s the only context I’ve heard burger used for specific to gay culture.