I think everyone who posted had the right answer--a nice feature of these fine boards!
I'll just say that:
1) The scene in GFII when the boy Vito is identified by a non-Italian-speaking immigration officer as "Vito Corleone" is absolutely historically correct. As I've posted elsewhere: my grandfather had a polysyllabic last name. When he came to Ellis Island, the immigration officer heard only the first syllable--and that's been our family name ever since.
2. Yes, it would be in Vito's interest not to change his name to "Andolini," in case Don Ciccio or his men were still looking for him.