Is there a term for "PhD at a department" vs "PhD at an (independent) research institute"?
I'm involved, as adviser, in a PhD program that falls exactly in your second category. The program, which is briefly presented here and here, is jointly run by my university (POLITO) and by the Italian national metrology institute (INRIM), which is a governement institute. Half of the available positions are paid by POLITO and half by INRIM. Frequently students are co-advised by advisers from both entities (this is not mandatory, but it's useful; and the student list on both sites is not up-to-date for what concerns co-advising information). 

However, the degree is awarded by the university only, and there is no formal distinction between the students that get their PhD from the above program and those who get their PhD from a program associated to a single department within the university. I'm aware of a couple of other similar programs in my field around Europe (here one in Germany), but I've never heard of any specific term distinguishing them.

Different is the case of a double degree which is awarded by two universities.