- there's
no rule that a museum is supposed to be publicly owned.
      
 I do not know about other states, but in New York - yes, real
 incorporated museums are public, officially part of University of the
 State of New York, and must follow the Rules of the Regents.
    
 
I think the keyword is incorporate.
There are private museums in New York. Whether they're incorporated is
another question - they certainly not bothered about getting tax-exempt
status for donations when they're able to afford the collection in the
first place. The owners might be financially solvent to not worry - and
then some.
But they are museums nonetheless - not just some ragtime outfit.
=Dan