After the whole York thing, I no longer feel all that apologetic about being precise about universities' names. Few random people encountered in the wild get the York thing. I went to the University of York, not York University. The first is in the UK, the second is in North York, which is in greater Toronto.
In Toronto, everyone assumed that I must mean York University.* Everywhere else, no matter how carefully I articulate it, they hear "the University of New York". I'm American, so that must be correct, right?
* The graduate student directory in my department at U of T had our previous universities listed on it. Mine was the only one for which the directory editor thought it necessary to specify country.