Communism is everything owned by the state.
Socialism is the means of production owned by the state.
Fascism (as understood by Mussolini) is a merger of state and corporate ownership for nationalist ends.
Oligarchy is ownership by the few, and democracy ownership by the majority.
Free markets are defined through ownership by The People, starting with self ownership and expanded to everything that logically follows from that.
"Community", "the collective", "public"... Can never be "the people" in any true and meaningful way. They're euphemisms for things like the state, which can never truly be "the people"/individuals, and is nearly always opposed to them overall by it's very nature.
Even if you believe the state is necessary (personally, I do), it has never been and could never be synonymous with the people. That's pure fantasy promoted by utopians dreaming of a world that doesn't and can't exist.
Mussolini once said “Fascism should more properly be called corporatism because it is the merger of state and corporate power.”... and that's how I think of it.