Approximately, a business that's open to the public is considered to be a public space, even if it's on private property.

A restaurant dining room is public, but the kitchen would be private (IIUC). A private club (ie you register for membership and only members are allowed inside) is private. (Aside: That's how indoor smoking establishments like hookah bars get around anti-smoking laws in Washington state.)

This leaves me uncertain about gyms - that seems more like a country club than a restaurant or hotel?