I agree with you, but I think it's safe to say that contemporaries didn't think that Rome had fallen at all. As you say the western emperor felt about 400, but the Eastern lasted to 1453. Even not taking the east into consideration, the Papacy was seen as a continuation of the empire, and it was the Pope who crowned Charlemagne emperor about 800AD. Circa one hundred years later, the Holy Roman Empire was formed claiming "traslatio imperii" (that is, that there was a legitimate line of succession since the last west emperor), and it lasted until the XIX century.