Earlier quoted context omitted.
Carl Zeiss Meditec /= Carl Zeiss. $AFX is the publicly traded non-semis division. Carl Zeiss AG is the parent company which is private
Correct. Funny enough though, their corporate structure and the name AG means they do have stocks, but they are not traded and 100% privately owned. For some reason I see this often with German companies, e.g. the German railway. Not sure why that is, although for the railway plausible since they are owned by the state that might eventually want to sell parts of it.
There is also SE which is a EU form for an AG, and various "partnership" forms that involve a partner that's fully liable. Usually, that partner is not an actual person but a "legal person", i.e. another SE or GmbH.
Even if you're not listed on a stock market, you might want to take on investments, e.g. "give me 10 million for 5% of the company" and I assume the latter is much easier with an AG.