Are small towns "rural" or "suburban"?
Rural, I would think -- though personally I'd argue that they should be a category of their own.
The Census Bureau identifies two types of urban areas
Urbanized Areas (UAs) of 50,000 or more people
Urban Clusters (UCs) of at least 2,500 and less than 50,000 people.
“Rural” encompasses all population, housing, and territory not included within an urban area.
So urban clusters (=small towns) are not rural, since they are a form of urban areas.
In Europe small towns are urban, I would think that was also the case in the US. It makes little sense to call them suburban.
In the US urban has the sense of being a part of a larger metropolitan area. The Census definition of UC recognizes that there are pockets in rural areas that otherwise meet the technical definitions of urban. As I noted in my post earlier in the thread UC's that are close enough to provide significant commuters to a larger urban center could be exurbs if there's a lot of growth happening around them, but otherwise fit as rural in the urban/suburban/rural divide.