re #18:
FWIW, at some point SUNY Binghamton deliberately rebranded to call itself “Binghamton University”. I read that this happened in 1992. To the extent that this name change “took”, it could be deemed now to have a private-sounding name. Though fooling nobody.
Regardless of whether it started out private, it was “SUNY Binghamton” for long enough that nobody even as old as I am knew it as anything else.