What’s the difference between “baby lamb” and “lamb”

For the times I've done gyros as such, you basically make a 'paste' out of it in the food processor.

Throw ground lamb in and give it a whirl, mix your spices and such in, put it on a spit, fridge to give a bit hard, and then on the spit. It comes out with a texture like that. That's how that kind is often done.

For 'proper' tasting meat (more like the one on the left), its alternating layers of razor thin meat and fat.

Brown the lamb on the stove top, or high temp oven. Then, braise rather than roasting. That means 2/3-3/4 of the pan is filled with liquid after you put the lamb in first. For flavorful liquids, consider beef, chicken or veg stock. In Malaysia I see a lot of grape juice substitutions for wine in traditional Italian recipes that call for braising. But I would dilute it, so it's not overly sweet. Perhaps some tomato juice would work as well.