I go to a well respected University of X. However, I actually attend a campus that is smaller and separate from the main large campus. As I look at graduate schools and begin drafting my statement of purpose, should I specifically mention (every time) that I go to U of X, City campus, or just mention the university as a whole?
If the rules and regulations are the same at both campuses, I doubt that you need to make a distinction. This is especially true if you would have been welcome to take classes at either campus without distinction.
But the university has an official name. If the satellite campus has a different official name you could, and maybe should, use that. But the "expanded" name may just be a geographic signifier without any official distinction other than geography. But you can ask the university for advice, of course.
My former university has several campuses in various cities in the same geographic region of the state. There is no distinction in the degrees offered and it is the university as a whole that grants them. The diploma shows no distinction. Is that the case at your institution?