Not all "public libraries" are paid for by taxpayers. Many communities have libraries that are paid for by donations, grants, or bequests and would be classified as a foundation.
Interesting. I don't know off hand if the ALA has control over them. In the future, I'll keep an eye on the distinction. I don't think I've come across it yet. Perhaps they don't have the same ALA-engendered problems public libraries do?