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.
I got slightly side-tracked when you said "we mean" and I wanted to clarify if you meant "the people having this discussion in this thread".
But I actually made the point:
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.
Maybe I could have added for more clarity:
And that's what the person you replied to means with "very few people have private health care in Europe, at least in the UK".
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.