As a non-American, I’m very confused by this. If it’s a town, it’s not rural by definition. Because, you-know, it’s urban.
Also, could we get a definition of town vs small town. Do you not have the concept of a village? (Village in the UK would be a settlement with a population of a couple of thousand, with usually a pub, local shop, maybe a post office and primary school if you’re lucky).
(edit: I was unaware of how prevalent legal village usage is in NY, but here’s my original comment)
I don’t think anyone really uses the term “village” in the NE unless it already exists as the specific name of the municipality or neighborhood (or they’re being cheeky). Maybe I’m too far into the metro-area suburbs, but not one village I know would classify as a village by OP’s definition. I don’t think Americans believe they have villages because they picture 3rd world huts, medieval towns, or eastern European towns with dirt roads.