From private company yes, but with private we mean the 'luxury' addon...the obligatory is regulated by the state, but administered by private company's.
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.