a political and economic theory of social organization which advocates that the means of production, distribution, and exchange should be owned or regulated by the community as a whole

The community is normal understood as the complete body politic.

A partnership is not socialistic. Lawyers are capitalistic in most books, but they have partnerships.

The organization of a business doesn’t matter in capitalism. Capitalism, boiled down to private property and market. A co-op competing with other companies largely without price control is just as capitalistic as IBM or Google.