Several trees and shrubs are commonly called 'bay' of one kind or another but culinary bay is Laurus nobilis. There is only one species although I have seen golden leaved varieties. Since bay thrives outdoors in much of the UK I should think it would be a cinch in SC. And I expect you already realised it will grow into a tree didn't you? Although it takes very well to topiary and clipping.

Here is a link that might be useful: My next door neighbour's bay tree

Are you really in zone 9 in Florence? I'd have thought you'd be in a cooler zone than that!