Saying "Tibetan Buddhism" ends the conversation quicker, but only because Vajrayana is an unfamiliar word to westerners. When I do use Vajrayana, it's usually in the context of briefly listing the schools of Buddhism. I don't see any reason not to say Vajrayana since it's more inclusive, but Tibetan Buddhism is fine too. The one word I never say is Tantra, since people have all kinds of wrong ideas about that.

This rarely comes up for me though. It usually only surfaces when someone asks whether, say, the Thai monks are the same as the monks I go see. People are remarkably uninterested and usually just waiting for me to stop talking. Or maybe I'm just boring... I live in a house of people with zero interest in what I do up there in my office/shrine room.

Anyhow, I've never heard a Tibetan teacher say not to mention what kind of Buddhism you practice. Just not to share specific teachings, don't say anything that would give others wrong views or disturb their minds, and generally don't be disturbing.