Please look at the Wikipedia map of Vatican City, where you will see that Rome is on all sides of VC, which really is an enclave. In this map, to the west of VC, is Via Angelo Emo; if you check the location of this road in e.g. Google Earth, you'll find that it is also in Rome. I'm really not sure what other evidence you need to convince you that you and your maps are mistaken. It might relieve you to know that the hill on which the Vatican stands is not one of the original 7 hills of Rome. Does this help?