Not really. A city is an administrative area, yes, but depending on where you are in the country the role you are describing might be a "township." In addition, there are often qualifiers to be a "city" in most states. Most cities end before reaching rural areas and you get into a township or "just a" county; we generally refer to "rural" as not in a city in casual conversation. Urban is different and not necessarily the direct antonym of rural; there's a middle ground.

California seems to be an exception where my GPS will report a city quite far away as "where I am now" when I'm in the sticks, but that might be my GPS. California is also less particular about administrative division naming than, say, Michigan.

City, town, village, county, parish, and township all have particular meanings that vary across the country. This is not an area where one can say "the USA" and be correct, as with many things in our really-50-countries patchwork. One example is that Wisconsin dubs "towns" what Michigan calls "civil townships."

America is confusing, even before you get into the debate over New Urbanism.