I was indeed refering to Charlemagne's kingdom and the later "Holy Roman empire" of Otto.But you are right, i don't consider them true successors to the western empire, i just used the term for convenience's sake.
It's been a while since i read that passage, but i believe that when Liutprand mentioned the "failure" of the Byzantines to protect Rome, he was talking about the 8th c. when the Pope turned to Pepin and his Franks for protection from the Lombards, instead of the Byzantines.
In any case, this wasnt my point. The reason i mentioned all these is to show the importance the Byzantines gave to the fact that they were the continuation of the Roman empire.What i'm interested in, is to get a picture of how they themselves(the Byzantines) saw,thought,felt,wrote about, etc. about the empire when it was still pagan and the capital located at Rome.