> It is socially impolite to yawn in public.

No, it isn't. It can be socially impolite to yawn unexcused, when someone is talking to you, as it has come to be interpreted as boredom rather than tiredness or similar. But it isn't inherently impolite to, for instance, yawn when walking down the street, or in a setting where someone isn't talking to you.

In my (limited) experience it is quite culturally dependent.

What you describe is in my opinion true for western cultures. In Brazil they are not so relaxed about it. Asia even less so.