The one country - Italy pass should be cheaper than the global pass, perhaps you are not comparing the same type?

To answer your question, there is no restriction on how and where you use a global pass, you would be free to use it in one country if you want to.

But there are mandatory seat reservations (3€ Intercity, 10€ Frecciarossa) with the Interrail/Eurail pass : Global or One-country while the Pass sold by Trenitalia might include them

The Trenitalia pass includes reservations but works with travels instead of travel days. As soon as you switch trains with the Trenitalia pass you use a new travel.