If you're not involved in adversarial contract authoring, I don't get the appeal of the "what about this edge case" game.
> Would the definition be based on possessing or owning?
Given that nobody calls people renting apartments homeless, I think this self-answers.
In any case, voluntarily being temporarily in a situation wherein one does not own property in not commonly confused with living on the street for reasons I believe to be fairly obvious. It has to do with that word "voluntarily".