Lamb Loin Chop is the traditional name in United States. Outside that tradition, butchers carry comparable lamb cuts under different names — sometimes the same anatomical piece, sometimes a close cousin. The alternatives below are grouped by country so you can match what your local butcher actually carries.
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.