Man that is really selling it. I lived in a car and it was my house. I will meet you in the middle there. But a tent is not a home. Even by my liberal standards of calling a vehicle a home.
I think the best term that has the most impact is Sleeping Rough Outdoors. You read that and it does not matter what side you are one, you now how the person is living and it describes the plight that many homeless face when just needing to get some rest.
The concept you are describing is a "homebum" which describes a person who sticks to one area. This is different than a "hobo" who keeps moving and is not as much a nuisance of homeless. A "rubber tramp" is a homeless person who lives in a vehicle.
I think part of the linguistic shift is to imply social ownership. Unhoused describes both their state, but also a failure of someone who is responsible for housing them.
In this case a use is being pushed by certain outlets that operate in lockstep. I'm not saying that's bad, but it's silly to pretend that it's an organic change.
These people are your neighbors, your neighborhood is home to them, they just don't have houses.