In general, "in" refers to a general area, whereas "at" refers to a specific point.

"in the centre" means "in the central area" -  "im Stadzentrum"

The restaurant is in the city centre.

"at the centre" means "at the centremost point" - "im genauen Mittelpunkt der Stadt"

The restaurant is right at the centre of the city.

mitten im Stadtzentrum is "in the middle of the city centre" (in British English, at least - Americans might use a different expression)

Traditionally if an English town has a cathedral or a university, it's a city.

There are places with (suffragan) bishops that do not have cathedrals, but they are not cities (e.g. Sherbourne, Ramsbury), so the presence of a bishop is not relevant. (I'm not sure if there are places which have a cathedral, but no bishop.) And it's a cathedral or a university, not and. There are numerous places that have a cathedral but no university that are cities. Including two of the smallest, St David's in Wales (population 1,800) and Wells (population 10,500). Compared to these, Cambridge is definitely a big city (population 125,000).

However, there are in fact towns which have a cathedral but do not have city status. And I think that with the conversion of polytechnics to universities, several large towns acquired a university overnight but did not automatically become cities. Contrary to the traditional definition, it actually requires letters patent issued by the monarch to be a city in the UK. It's all explained at https://en.wikipedia.org/wiki/City_status_in_...