Earlier quoted context omitted.
What is private/semi-private in that phrase is the hospital room that the insurance covers, not the insurance itself.
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.