I'm not Italian, but my impression from traveling in Italy is that every town\village\grandma has their own version of nearly every Italian dish and they all consider theirs to be the traditional version. I think I got told three or four times that I was in the village where focaccia was invented - different villages, that is. And they were all quite different focaccias so I guess they were all right.

Side note: bechamel lasagne is usually called lasagne al forno and I think it comes from Bologna. In many other parts of Italy they don't use bechamel in their traditional lasagnes, and they often do use ricotta and mozzarella.