My main problem is that you don't know that unhoused, which is a term I use in this field, is not the same as sleeping rough, another term I use in the field. Sleeping rough is what you might call sleeping on the street; it's the stereotypical dossing on a park bench or in a cardboard box under a bridge, etc.
Unhoused is the umbrella term, including rough sleepers, people in their cars, people couch surfing, people in hotels that they can't afford for very long, etc.
I think you're right that it's a cultural thing; here in Australia we'd say unhoused (or homeless) and rough sleeping without anyone blinking an eye.
What do you call it in your neck of the woods when someone is both unhoused, ie without a home, and sleeping in a public place, vulnerable to strangers and the elements?