A homeowner association is a private, legally-incorporated organization that governs a housing community, collects dues, and sets rules for its residents.
https://en.wikipedia.org/wiki/Public_institution_(United_Sta...
A public institution is a juristic person in the United States which is backed through public funds and controlled by the state.
HOAs are not controlled or funded by the state, no more than Private Businesses are. municipalities are controlled by the state.
if municipalities have the same reach into how people manage their homes like HOAs do then the distinction may not matter. it is however a legally significant distinction.
one point where the distinction matters is that HOAs can limit your freedom of speech, while municipalities can not.
https://www.hopb.co/blog/freedom-in-associations-exercising-...