Web 2.0 isn't "community-coined". The term was invented by Tim O'Reilly to sell conference registrations and books. It's no more of a "brand name" and no more nebulous now than it was when he coined it.
AJAX was coined by Jesse James Garrett. I'm not sure that he would consider himself to be a community.
Yep, that's what I meant. AJAX was "community coined" in the sense that it was coined a member of the tech community, not by a company (as with a trademark or commercial brand name) or a standards body (as with official standards like "ECMAScript").
It's like when "the neighborhood" forms a "neighborhood watch." Not everyone in the neighborhood spontaneously starts it - it's usually one or a few people who come up with the idea and then everyone gels around it. Or like how Java was a technology created by Sun, even though it was really Patrick Naughton - it's a synecdoche of sorts.
All that said, while the above stands for AJAX, you probably have a point regarding "Web 2.0" since it was tied to O'Reilly as a company via trademark, though I'd argue that the community essentially "stole" this brand and made it a lot of things that it never originally was.
AJAX was coined by Jesse James Garrett. I'm not sure that he would consider himself to be a community.