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).
American’s rarely use the word “village” - a “town” can be anything from a few thousand people down to almost empty. It’s very subjective - some places that make a proud point of calling themselves “the city of …” are IMO small towns at best. Rural in America means surrounded by a fair amount of countryside, be it farmland or nature. I think most Americans loosely associate “urban” with closely packed tallish buildings,maybe 5-7 stories or more, and mostly wouldn’t agree with you that a “town” is urban by definition. In common speech there’s no clear threshold between large town and small city.