Do you think Cornell defining homeless but not unhoused might be a hint that they are synonyms?

That is quoted US statute, made available in an easy to access format through Cornell, not Cornell defining anything.

You gave me a definition of homelessness, which doesn’t counter what I said in the least

I gave you an article that discusses the terminology and how it is used for context that differing terminology is no inherently all different names for the same thing. It doesn't define anything, it just makes it clear that there can be differing terminology that means different things and that the whole thing is a complicated topic. That is why I linked the article, not to prove definitions that don't exist because the terminology varies in usage and consideration of importance.

But I guess sarcasm is easier then admitting you are wrong.

Any statement of how words are used will be wrong somewhere, except for things like the quoted law that is true in the context of written law in that country/region/whatever. There is always local or regional differences in usage.

So I am right about how we use it in our context to explain the concept of homelessness in the legal context even if some other people think it is a synonym, but thing other terminology has an important distinction. That is what I said, and if you can't understand there isn't a black and white defined terminology for all the variation then you aren't getting my point.