[Merry]

I think it's a bit like the differences between idiots, lunatics and imbeciles on the census - there was an official difference, but that wouldn't have to be the same as what people took the words to mean.

I would think an inmate was someone in an institution because they were either a permanent resident because of infirmity or illness or temporary because of temporary illness or because of financial destitution. An inmate might also be in a prison. In any case, they were probably there because someone else decided they should be.

A patient sounds like someone paying their own way for the care received be it temporary or permanent. They would also be able to discharge themselves from the place giving them care.

I would guess there's a huge cross over in the use of these terms.