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).
Depending on the state there may be different formal definitions based on population and incorporation status. In Ohio, we have townships, villages, and cities. In Pennsylvania they have “boroughs” instead of villages. In NY a borough is a subdivision of a city. I don’t think they have the township organizational structure in Vermont. In Maine there are unincorporated territories (usually just numbered).
The role of the county government can vary significantly from state to state too.
My understanding is that in Virginia the city/town that serves as the county seat is explicitly not a part of the county it’s located in!