You are absolutely right that there are degrees of homelessness, and I would argue that the only people who are 0% homeless, not homeless at all, are people who own their homes outright. Which, yes, means that almost everybody besides the richest of the rich, since at least feudalism onward, has been homeless to some degree. If you're mortgaging your home, the lender has an interest in it and can take it from you so you don't really own a home, you're borrowing someone else's. If you're a tenant renting the land -- whether in cash as renters today or in kind as tenant farmers in the feudal era -- then you are explicitly borrowing someone else's land, and don't have a home of your own. In a sense, even people who "own their homes outright" in the usual way (fee simple) are still technically, legally, tenants on the State's land, which is how property taxes are justified.
And in a very real and intuitive way, unless you really explicitly own something outright, you don't really "have" it. Say you drive a beaten up old clunker normally, and it's broken down again, and your rich friendly neighbor lets you drive his fancy Ferrari to the store for groceries. While there, you run into an old friend you haven't seen in years, who sees the car you're driving and say "Whoa, man! You have a Ferrari now!?" Is the correct answer to his question "yes" or "no"? If you say "yes", he will think that you own the Ferrari, so "you have a Ferrari" means "you own a Ferrari". If you don't own it, the answer he will expects will be "no" and an explanation of why you're driving one anyway.
In that same sense, "do you have a home?" means "do you own a home?", so if you don't own a home, you don't have a home, and not having a home makes you homeless. Homeless people can borrow other people's homes, to various degrees -- a friend letting you sleep on his apartment floor is different from a richer friend letting you sleep in his summer house, and renting or mortgaging something is just yet another degree of borrowing someone else's home -- but they're still essentially homeless.