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).
It’s not rly a term that’s used but it’s the correct term as well as what it says on welcome signs. Like “welcome to the village of [name]”. Most people just use the term town/small town. I was just explaining the true difference between the three for OPs question