Is there a term for "PhD at a department" vs "PhD at an (independent) research institute"? All PhD students are getting their PhD from a university (as far as I know). 


But some PhD students follow a PhD program that is just a standard program at their department (e.g. department of math, or physics, or sociology), do research under a supervisor who is just affiliated with the department. The supervisor may also be affiliated with research institutes, but the relation between the PhD student and the supervisor is not created via, or dependent on, or related to those research institutes: the relation is solely based on the fact that the PhD student and supervisor are affiliated with the same department. 
Then there are PhD students, who (though they are of course also affiliated to a department) are enrolled into a PhD position that is specifically affiliated with a research institute. For example, there may be a research institute that is affiliated with university X, all researchers there are affiliated with X, but it is a somewhat separate institute. The PhD student would be specifically linked to that institute, rather than merely to his department. 


Are there generally accepted terms that distinguish between these two types of PhD positions?
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.