One of the conditions of citizenship is renouncing previous citizenship, so I was wrong in my previous answer.
Italy would have no claim on you.
Born here of citizen parents is the standard.
Interestingly, Ronald Reagan was only one generation removed from being able to get an Irish passport. Under their law, citizenship can pass down as far as a grandchild, and Reagan was the great-grandchild of an Irish immigrant who become a citizen only after Reagan's grandfather was born.
Have to disagree, there are only two types of citizens, natural born and naturalized. I see nothing in the constitution or any US law defining any other type of citizen.