>I haven't been to Berlin, so I don't know exactly how big it is, but one would not typically call "within 2km of city center" a suburb. One would typically call 2km from a city center, "the next neighborhood over in the city".
This is an American/Canadian vs Australian (and possibly European) thing.
From speaking to Australians, Downtown is the Central Business District, and everything else is a suburb.
I'm not sure where one would place the "center" of Berlin, but Berlin-Mitte (which you could translate as "Berlin Central") is really mostly banks and insurance companies, tourists and expensive hotels. It's not very urban, or alive. It's just... there.
This is an American/Canadian vs Australian (and possibly European) thing.
From speaking to Australians, Downtown is the Central Business District, and everything else is a suburb.