Westside lifters started doing belt squats in 1975. I personally realized in the early 1970s that my quads were somewhat smaller than before I started doing box squats. Of course I Olympic lifted first and used a close stance with a raised heel. But box squatting with a wide stance while pushing the knees out to the sides placed most of the work on the hips and glutes. It was at this point that I started to belt squat, including standing on a ramp and not locking out the legs to keep tension on the quads.
Westside lifters would belt squat after box squatting or on max effort day after a good morning or a deadlift of some type. Today, we use several variations of the belt squat. Variation include belt squatting on a box; belt squatting without a box; walking on the belt squat platform until failure; walking forward where the cable is behind you, bent over like deadlifting (this is an unreal glute developer); and walking backward with tension on the front of the legs. We also do a lot of calf work in the belt squat machine.