It says the "the" is needed, and in the U.S., it is. However, I know that in the U.K., they say that someone is "at university." Is that a different meaning, or should both be accepted?
According to this source http://www.learnersdictionary.com/qa/at-university-or-at-a-university "at university" and "at a/the university" mean different things. "At university" stands for "attending university" and "at a/the university" describes the location. Without a context "в університеті" describes the location. For "attending" I would say "вчитися в університеті". So, "at a/the university" is closer to the meaning of "в університеті".
I'm likely not considering all possible scenarios, but generally, "the" in English isn't necessary. I've heard- and used- both of them myself in conversation with all sorts of folk.
I am "at university" is the same as "I am attending University" . I am "at the university" is the same "I'm at university". The latter is lazy grammar and implies a knowledge of the person attending University, but this type of speech is common.
Yes, in British English when talking to friends we often just say "at university/uni" when talking about location.
"Where are you?" "At uni." is something that can be very commonly heard here. We sometimes say "at the university" too, but it sounds more formal and is more likely to be used with strangers than someone you know well.