I think that the rack of lamb is usually cooked with the fatty side up so the juices flow over the meat while cooking. Perhaps cooking them vertical is about letting the fat run off. That's just a guess. I always trim off as much fat as possible from all of the meats I cook but the rack of lamb takes a considerable effort to do that with until after it's cooked.
In the Mid-East, it is popular to cook lamb over an open fire. A charcoal brizier serves well. the trimmed fat is thrown onto the hot coals and the aroma is magical. as the fat is being melted on the coals, the meat is put on the racks. The smell of the fat, and no calories, is absorbed into the cooking meat.
I cook rack of lamb on the charcoal grill. I cook it fat side down at first, until some of the fat melts off the rest gets slightly charred, then turn it fat side up to let the remaining fat baste the meat a little, and finally stand the roast with the ribs pointing up and finish it that way. You have to watch it closely so that the melting fat does not start a grease fire. Grill it with the cover on and the vents partly closed to keep it from flaring up. Almost any herb mix works, but I prefer to rub the meat with olive oil and then sprinkle chopped rosemary on it. I use fresh rosemary, strip the leaves off and soak the stems in water, and then throw the wet stems on the coals to provide some smoke flavor.
