I don't think I've ever heard anyone say "private insurance" to refer to the "supplementary insurance" (Zusatzversicherung / assurance complémentaire / assicurazione complementare), so if it's used in that sense it's not universal.
Supplementary insurance is conceptually different, by definition, from comprehensive private health insurance which provides alternative coverage for what is already covered within the public healthcare system.
Not just the room, also better treatment more expensive medication etc
That is whats called private, if you never heard the phrase private krankenkasse...well everyone uses it, it's also the first (maybe second) question when you go into any hospital.
Not just the kind of hospital room, but I'd say it's the main difference (there are many different supplemental insurance products anyway). And I don't thing "semi-private insurance" is a thing anywhere else.
Even if you're right and the term is used more generally than I thought, my main point stands: it's quite different from what people in other countries call "private health insurance" so in the context of the discussion here the distinction is relevant.