I just checked some Card Agreements:

-Chase Visa Signature covers Italy.
-Citi Mastercard - I could not find Card Agreements reg. car rentals in my file, never got them,
-American Express Gold and Starwood (both private and business cards). I could not find that any countries are excluded(!?!). However, I know for sure, that I have read some years ago that some countries, like Italy are excluded.
Am I blind? Can somebody point to the paragraph please.

BTW, regarding primary/secondary, Chase Visa is pretty clear: PRIMARY for USA, SECONDARY for abroad.