From an American perspective!
I think that the structure of Italian towns doesn't quite correspond to the structure of American towns; so an answer from BrE might be more appropriate. Nonetheless, I thought I'd contribute.
My understanding is that all of Italy is divided into a municipality (a comune), which contains the actual town and any outlying areas.
However, that's not generally the case in the United States, especially out west.
Take a look at Yakima, Washington, a town in my state.
The red area is the city of Yakima (in Washington, there is no legal difference between a "city" and a "town," although my understanding is that there is a difference, especially in New England), whereas the rest is Yakima County, somewhat equivalent to an Italian province, but still subordinate to the state of Washington.
So, while they share the same name, what you're discussing (the city or the county) is obvious.
So here come my suggestions:
- Usually people in these areas who live more rurally will talk about going "into town," where which town is usually obvious from context.
- The "city proper" refers to the legal boundaries of Yakima, as you can see, there are suburbs of Yakima on the map, and those are distinct legal entities. So, if you were discussing the area and needed to refer to something in the city of Yakima, you could use that, in contrast with something outside its legal boundaries.
- People who live in Yakima proper might tell people "I live in Yakima," if they don't live in the city boundaries, they would say "I live outside of Yakima." I have friends who don't live in a city or a town, so they say "I live in Snohomish County."
Now, that's not to say some cities don't have rural areas within their boundaries. Anchorage, Alaska, comes to mind. Since it is a unified city-county, it contains huge areas of rural land, and Wikipedia calls the actual city part "the urban core," which would only work if the city in question has urban areas.
You might also try "downtown" to refer to the center of the city, even when it is spread out.
This is a long way of saying that, at least in my neck of the English woods, this is a problem that doesn't exist, I would be interested in hearing from some Brits or even some Anchoragites, but I hope I gave you some perspective.