Lamb Rack is the traditional name in United States. Elsewhere, butchers sell the same piece — or a close cousin — under a different name.
Same anatomical primal but younger animal — abbacchio is whole-animal-young, regular lamb rack comes from older lambs.
Anatomically adjacent — rack is the rib end (8 ribs), saddle is the loin section behind it. A whole rack-plus-saddle is sometimes called a *long saddle* or sold as a guard-of-honour preparation.