Lamb Loin Chop is the traditional name in United States. Elsewhere, butchers sell the same piece — or a close cousin — under a different name.
Same anatomical cut from the saddle, but Barnsley chop is cut across both loin eyes plus the spinal bone — a single chop spanning both sides of the animal — while a standard loin chop is cut from one side only.
Same anatomy: a saddle is both loins attached as one roast; a loin chop is a cross-cut single-loin slice from that same section. Saddle is the whole, chops are the divisions.