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?