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).
A village would be an “unincorporated town” in the US, administered under the county. There are a few actual incorporated villages, but they are small towns that just chose to call themselves that.
The main difference is that an incorporated city/town/etc has its own municiple government … a “village” would still be run by the county.