J.H. Walsh, quoted by Webster in the dictionary entry, is wrong. He states that the canter is an artificial gait. Mr. Walsh probably never saw a wild horse, let alone a whole herd of them. In fact, the canter is a gait natural to the horse, and I myself have seen wild horses who have been wild for many generations canter across the desert.
The walk is a four-beat gait: each hoof strikes the ground independently. The trot is a two-beat gait: the right front hoof and left back hoof strike together, then the left front and the right back, again together. Hence the trot is quite bouncy. The gallop is again, like the walk, a four-beat gait: each hoof strikes independently (and as discovered early in the history of photography, there is a point in the gallop where all four hooves are off the ground simultaneously).
The canter, however, is odd in that it is a three-beat gait. Two diagonal hooves strike together (right front and left back, or left front and right back) and the other two strike independently of the others and of each other. The overall speed is just short of a gallop, but the motion of the horse rocks from front to back with the striking of the independent front and back hooves, like a rocking horse. I wonder whether the toy was invented to mimic this gait.
Riding a cantering horse is, above everything else, fun. The speed is exhilarating but not in the slightest degree alarming, even for a novice. The gentle rocking is soothing, recalling childhood. Also, there is a lot of emotional connection, by necessity, between horse and rider, and horses do not canter when they are alarmed. When horses panic, which they do quite easily, being prey animals, they gallop. In the wild they canter when they want to get somewhere quickly, but are not particularly afraid of anything. Perhaps that comfort and ease is communicated to the rider by that mysterious emotional bond.
In Western riding the canter is call the lope, and the word is perfect. Easy running, without fear, without urgency, and with just the hint of a smile.