I always say 'town' for Manchester city centre, I thought that was a class thing...
But for how much longer eh? Manchester and Leeds, the actual areas that are the city, are undergoing huge gentrification. I know both places very well and they are unrecognisable now to what I knew as a young person. Trendy happening, thriving wealthy, like Burnham ATM I suppose
But its the poorer parts or GM, and (W,Yorks re Leeds) that get forgotten. Places like Rochdale, Wigan, Oldham etc. These are large towns, together larger than Manchester by population, with their own identity. They get overshadowed
Trafford and Tameside are invented places really, but the towns are their thing