The "public" is private anyway, as the Krankenkassen are all private companies (although strictly regulated by BMG).
But yes, there's concept of public (statutory) and private (voluntary) insurance plans.
On top of that, as you said, most (all?) Arztpraxen are also private entities. Same goes for hospitals (I guess excluding places like universities and Bundeswehr).
Many hospitals are part of a municipality or a university (again, established by public law), many others are organized as private companies (either publicly or privately owned). Non-hospital doctors are almost completely private entities.
This is not so clear-cut as in other countries.
Translated from Wiki:
> As a public corporation with self-administration, a health insurance fund regulates its budget on its own responsibility. In doing so, it must fulfil legislative performance requirements (compulsory benefits) and may in some cases go beyond this (statutory benefits). According to § 260 para. 2 SGB V, its operating funds should not exceed one monthly expenditure.
Health care might be organized differently than in most other countries but that does not imply that those institution are private.
Is the City of Munich also a private company? It has self-administration, is responsible for its own budget which can't be negative, has to operate within legislative bounds including giving compulsory benefits to its residents (which are exactly the criteria you quote). Of course not, that's a city. What about the Technical University of Munich? They even have "members" instead of residents in addition to the things above. It's all the exact same kind of legal entity. Saying one is public and the other one private doesn't make any sense. What's the difference between those in your eyes?