I'm not sure. The wording of the marketing language is ambiguous:
...could mean:
A) There is no coverage for personal rentals
B) These is secondary coverage for personal rentals
The difference between primary and secondary is immaterial when renting outside the USA. I hope the language in the benefits guide is more precise.
I have checked into this last year,
it was B). IIRC the benefit guide has more precise language.
When it is rented outside US, it does not make any difference as outside US the coverage is always primary, exactly as you point out.
Though personally I would not hide the fact that the car would be driven into Italy. The rental car company itself would have more specific information on which countries are allowed which are not.
You will get into SERIOUS trouble if the country you drive into is NOT allowed on the rental agreement.
Always pay close attention to that. When in doubt, get this thing clarified with the rental car company, not your insurance provider. Because if Italy is not allowed, then your insurance would be VOIDED due to you violate the rental agreement's T&Cs.
Cross border rentals when Italy is involved, can be a very messy matter. When we rented in Croatia we anticipated crossing Italy (Trieste) before we re-enter Slovenia/Croatia. There were some discussion in the Avis board. Some folks Italian rental could not enter Croatia. They had to park their Italian rental in Italy side, flew to Croatia and rented another car in Croatia. Something like that.
Fortunately for us Avis rental agreement did not prohibit driving into Italy, but did include some Eastern Europe countries that were not allowed. We also needed to buy additional insurance, IIRC, was liability, for taking the car into Montenegro. The border crossing officials actually examined the "green book" Avis supplied to us. Though you could also buy the insurance right at the border, actually slightly cheaper than using Avis' own, but I would think in case you need to use the coverage, dealing with Avis' own would be much easier than dealing with some vendor you bought from at the border crossing!