It's that sort of nonsense that has facilitated so many dishonest people to date .... can't pay means that to pay would result in the person not having enough money for basic food, light, heat and shelter for themselves or their immediate dependents. Anything more than that, is an optional luxury and does not prohibit ability to pay so it then becomes won't pay.
Well I am sorry you think I am talking nonsense. I think you are indulging in wishful thinking.
It would be great to be able to neatly separate those who "can't pay" from those who "wont pay". In reality that isn't possible. Instead of two groups there is a continuous spectrum.
Just from your points above, what is basic shelter. Should a family give up a luxurious home, to move to basic shelter. Who would provide that basic shelter, it might well cost more. Who are immediate dependants, adult children, adult children with special needs. What about transport costs, would the family have to stay all day within walking distance of their basic shelter. What if the live in the country.
These are not trivial questions, they are practical impossibilities to dividing "can't pay" from "wont pay"