What does the belly have to do with the fall of the feet? The gaits have to do with the pattern of the foot falls, only. Not the way they carry any part of their body. You CAN tell a rack from a trot by the motion of the tail. Up and down vs Side to side, but it takes some studying.
From my favorite book, here are some definitions of gaits:
The walk is a slow, flat footed, four beat gait.
The trot is a two beat gait, in which the diagonal fore and hind legs act together.
The pace . . . is a rapid, two beat gait in which the lateral fore and hind legs work together.
The amble is a lateral gait usually distinguished from the pace by being slower and morebroken in cadence.
The rack is a fast, flashy, lateral four beat gait. It was once defined by the discarded name "single foot"
The gallop is a fast, three beat gait, in which two diagonal legs are paired, their single beat falling between the successive beats of the other two legs.
The canter is a three beat gait done under restraint. Sequence is the same as the gallop.
The running walk is a slow single-foot or four beat gait, with the break in the impact or rhythm occurring between diagonal fore and hind feet. In the stepping pace, which is also a slow, four beat gait, the break in the impact occurrs between the lateral fore and rear feet.
The foxtrot is a short, broken, nodding, somewhat uncollected trot, a gait sometimes used as a substitute for the running walk.