Kuan, the lamb rack is the portion left after the shoulder,breast and loins are removed from the main carcass. We call that a hotel rack (same with veal) This is before the primal is broken down to any sub-primal cuts.
Last night I tried a boil in bag rack of lamb from Trader Joe's. It definitely had the breast still on. Four bones, no fat cap, two layers of breast muscle on top of the rib. Kinda had the texture of short rib. I don't think you can call that rack of lamb.
The lowered number ribs, 1, 2, 3, etc., are towards the front; or as you said, closer to the "shoulder." The higher numbered ribs, 11, 12, and 13 are towards the rear of the animal. In this case, the last rib, 13, butts up against the loin.
A lamb has 13 ribs. Ribs 1 through 4 are used as riblets. Ribs 5 - 12 are the rack. I don't remember how the 13th rib, as much a part of the loin as the rib, is used -- if at all.
If you order "rack of lamb" in a restaurant, you are nearly always served a half-rack -- that is to say, 2 double chops or 4 ribs.
The current "hot" presentation is to french the ribs before cooking, then remove every other rib at carving, so each double chop is presented with but one bone. Bones used to be protected during cooking, so as to present white; but that's no longer the style. Modern cooks seem to prefer them with a slight char.