According to this source ( https://www.newhampshire-demographics.com/cities_by_population ) most of these are inaccurate. Not a big issue would just make this quiz a lot bigger if it was either bigger or more accurate.
Derry, Londonderry, et al are definitely incorporated. The difference that they are towns, and not cities, is just a matter of slight differences in the style of local government. No one around here would think of them any differently than they would many of the cities you have here. It's not like out West where there are actual unincorporated areas with no government.
coming from NH, this quiz isn't totally accurate -- for example, Exeter and Durham are two municipalities that have >15,000 people. while they are technically towns, in NH there isn't really a difference because none of our cities are that populated