'In' is used to describe the exact location of where someone or something actually' is/was' or does/did something. I live 'in' a house. The TV is 'in' the sitting room. It is often used instead of 'inside'. Where's the sugar? It's in/inside the cupboard. We use 'at' for more general location. I study 'at' University (Usually without 'the' unless your're referring to one specific university.') is correct because you are not referring to your exact physical location.

"I study at university" , "I study at the university of..." or "I study at (the name) University" The rule is to use "at", just like when saying "I work at (name of the firm or institution)" rather than "in", in these cases.

AT at = at the place of at the university = the whole university and its systems

If I understand right, that I may use the phrase "I study at (the name) University", for example: I study at City University London. I study at Michigan State University. I study at the University of St. Andrew. but I can see that some write such, for example: (*) I study at the City University London. (**) I study at the Michigan State University. So you can see that the article "the" exists. Is this correct to write a name of the University with "the" in these cases of (*) and (**)?