'cathedral city' - but see explanation below
Traditionally, in British Englsih, a city was a town with a cathedral, and all the others were simply towns.
They are all towns, including (in English national eyes) the mother of them all, London Town. I think what most people understand as a city is one with government buildings, a university, maybe a cathedral still, theatres, cultural centres etc.
If a town 'only' serves as a shopping centre, workplace and central point for (primary) schools, local government and transport, it would not be called a city.
So some 'cities' are in fact small in terms of area and population, but have long cultural traditions, and some towns have grown large because of industry and good transport connections without ever developing the 'heart' that makes a true city.
If in doubt, I always write town, but some people occasionally feel I am underrating their 'city', so I try to find out what kind of conurbation it is whenever possible.
I don't think native speakers regard cities as 'finer' or better than towns. There are just different facilities - and a town may be far better and more convenient for shopping than a congested city with narrow streets and nowhere to park your car!
· n. (pl. cities)
1 a large town, in particular (Brit.) a town created a city by charter and containing a cathedral. Ø N. Amer. a municipal centre incorporated by the state or province.
2 (the City) the part of London governed by the Lord Mayor and the Corporation. Ø the financial and commercial institutions in this part of London.
– DERIVATIVES cityward adj. & adv. citywards adv.
– ORIGIN ME (orig. denoting a town): from OFr. cite, from L. civitas, from civis ‘citizen’.
1 a built-up area with a name, defined boundaries, and local government, that is larger than a village and generally smaller than a city. Ø Brit. dated the chief city or town of a region.
2 the central part of a neighbourhood, with its business or shopping area.
3 densely populated areas, especially as contrasted with the country or suburbs.
4 the permanent residents of a university town. Often contrasted with gown.
5 N. Amer. another term for township (in sense 3).
– PHRASES go to town informal do something thoroughly or enthusiastically. on the town informal enjoying the nightlife of a city or town.
– DERIVATIVES townish adj. townlet n. townward adj. & adv. townwards adv.
– ORIGIN OE tGn ‘enclosed piece of land, homestead, village’, of Gmc origin.
There are guaranteed other references and opinions, but that is my two penn'orth!
Concise Oxford Dictionary