The term "private company" has no relation to whether or not it is traded publicly. It simply means the company is not owned by or controlled by (in some sense) the government, but by "private" individuals
I guessed. I still don't understand why it is necessary to differentiate a category with 1.7 million members from a category with 17 members. Nobody here is ever, ever, ever (almost) discussing government corporations. I don't understand why people don't just say the 1st amendment doesn't apply to corporations. Everyone would know what they mean. Adding "private" suggests they are distinguishing it from something else. And maybe they are, but it always sounds like they're either misspeaking, confused, or repeating a meme.
Aren't most universities public companies even in USA? I know some of them are private, but there must be way more than 17 public ones. Add all the public schools, police departments etc, and you get quite a lot of them.