Turns out that the tea ceremony we know in USA (Sen no Rikyū, wabi-sabi, small tea huts, poetry of the simple and humble…) is wabi tea (wabi meaning simple as in (as far as I understand) “простолюдины”).
There’s also samurai tea, baku-cha ( deriving from work of Kobori Enshu ) (also growing from Rikyu, but claiming to be an improvement) and shōin, or drawing-room tea, for priests and aristocrats. This one seems to be served in large pavilions (e. g. Golden, Silver, Floating pavilions in Kyoto).
Of course, once one starts looking one finds that the samurai tea at least is being exported, that English-language practitioners of both are all over, and I just never met any and am, from ignorance, failing to recognize the distinction when I see someone drinking tea in popular culture. Since they influenced each other continuously the differences would be even harder to see for an uneducated person.
After all, there’s a whole long history of tea in Japan of which I was almost entirely unaware.
Still, one would think that shōin tea would be easier to tell apart from the other two and it takes serious digging before one even learns that it exists. But, since shoin reception halls are used in the samurai tea way perhaps it got enfolded into that to the point where, again, a profane viewer will not be able to tell them apart at a glance.