Liverpool is a city.
A village
It goes Hamlet - Smallest Village Town City CMA Megalopolis - Biggest
People who lived in the village built farms and houses. The village grew bigger and became a town and the town grew bigger and bigger and became a city.
Neither: a city.
A town and from the 16th century a city.
Yes, there is a village called Liverpool in Onondaga County, New York. It's near the city of Syracuse, NY.
City
A small community of citizens. Similar to a city or town, but with a smaller population.
town, village, city
village, city
Town, village
Tuscumbia, Alabama: City population: 7,856 Tuscumbia, Missouri: Village population: 218