There are different interpretations of the word "public". The parent I think uses it as a legal term, i.e., deviations can be reported to and handled by law enf

There are different interpretations of the word "public". The parent I think uses it as a legal term, i.e., deviations can be reported to and handled by law enforcement: e.g., in the US one can photograph anything and anyone on a public street, but usually cannot walk there naked. This is the definition I personally use, too. The meaning you seem to be using the term to mean welcoming everyone by choice of the owner (e.g., "everyone can come to my restaurant"). However, in this case the owner can also impose their own visitation rules (e.g., prohibiting a dress type that is offensive to one group but perfectly acceptable to another).