Oh, I assumed it was the other way around: I may have a roof over my head, a car, a couch, somewhere to be temporarily, but I don't have a home. So it's all homeless, and some homeless are also without any form of housing...

pretty much this. I volunteer with a couple of homeless outreaches and even we don't use the term "unhoused" because if you've spent any amount of time in a shelter it's the absolute last place you want to be. The problem is people need homes and so someone decided to kick the can down the road and say that the term homeless was demeaning and decided that unhoused was more politically correct all the why whomever came up with it I highly doubt spent a single night in a shelter.