By an even tighter definition, Franks are Germans in origin. To say that, however, would be so misleading as to negate history from the time of Charlemagne at least.
Location in Italy does not make everybody "Italian," just like location on the Balkans does not make disparate people their all "Balkans."
My definition of an Italian is fairly straightforward: a person native to Italy. Whether or not the Tyrol is Italy might be open to question. Whether or not Rome is, is not. I agree that the north western part of what we now call Italy was then Cisalpine Gaul and southern Italy and Sicily was Magna Grecia, but Rome was always Italian. What occured from about 300 BC to 500 AD was the Romanization of much of Europe including all of what today is Italy. Roman wasn’t italianized, Italy was romanized.