A canter is a three beat gait with leads. The foot fall pattern is hindfoot, diagonals together and remaining forefoot. For the left lead, the footfall pattern is RH, LH-RF, LF. For the right lead the pattern is LH, RH-LF, RF.
The gallop is a four beat gait with the same footfall order, but the diagonals are disunited giving us: RH, LH, RF, LF or LH, RH, LF, RF.
When a horse is taught to canter in place or nearly in place, the paired diagonal does not need to reach as far and the canter can become a 2 beat gait where the horse bounces between the forelimbs and the hinds limbs - RH-LH, RF-LF.
A very collected canter can also become a four beat gait with a moment of suspension that resembles a two beat gait. For the left lead this looks like - (close together) RH,LH, Pause, (close together) RF, LF.
Collected canters are advantageous in close combat situations. They are less useful when galloping down a lyst in the joust or chasing a running stag. Artists may use the two beat posture to emphasis the collection, such as in fectbuchs.
The two beat posture may have become a stylized way to position a running quadruped. Note that in the hunting print, the galloping horses, dogs and stags all display the same gait. It is unlikely that the dogs and stag had been trained to perform a highly collected canter.
Although horses in harness are capable of highly collected canters, those gaits generate more upward motion than forward motion. A horse in draft needs forward motion to maintain vehicle movement. The movement detailed on the Pharaoh's Paint Box with the large stride likely alludes to a gallop motion as opposed to a collected canter that might be used in ridden combat.
BEAT 1
Start of the Stride (Big Canter) - Outside hind on the ground. Diagonal pair (inside hind and outside fore) advancing
BEAT 2
Outside hind and diagonal pair on the ground. Inside fore moving forward.
BEAT 3
Completing the stride - Inside front landing.
START
Bending the joints of the hind leg and just starting to move the diagonal pair forward. Compared to the Beat 1 photo, this image is earlier in the point of the stride and the horse is taking smaller steps.
Period Descriptions of Canters
Xenophon (~400BC)
In "Three Essays - On the Duties of a Cavalry General, on Horsemanship, and on Hunting" Xenophon (~400BC; trans Dakyns, 1897) references a footfall pattern more in keeping with modern notions of the canter:
the preference is given to starting on the left foot, it will best conduce to that lead if any, while the horse is still trotting, the signal to gallop should be given at the instant of making a step with his right foot. As he is on the point of lifting his left foot he will start upon it, while turning left will simultaneously make the first bound of the gallop, since as a matter of instinct, the horse, being turned to the right with his left
William Fitz-Stephens (1170)
In "A Description of the City of London" William Fitz-Stephens (1170) paints a picture that matches much of the art:
...their gallop (in which their two sets of legs, front and rear, are thrust out forwards and backwards, in opposition to each other).
De Reayo (1548)
In "Doctrina del arte de la cavalleria" (1548; trans Fallows), de Reayo expects jousting horses to gallop forward and freely:
If you should have occasion to joust. it behooves you to have a horse that can run freely and that will gallop.
Claudio Corte 1562
Many Masters, including Corte in "Il Cavallarizzo" (1562), invoke music to describe the desired rhythm of the canter:
Much is to be commended Sir Cesare Fiaschi truly honored knight, who has put the above canter to music, to communicate clearly and well (as I understand it, the big time) and the big measure that is requires.
Il cavallarizzo Claudio Corte 1562 (Tomassini translation)
Blundeville (1561)
In "A Newe Booke Containing The Arte of Ryding, and Breaking Greate Horses Together with the Shapes and Figures Of Many and Diverse Kyndes of Byttes, Mete to Serve Diverse Mouthes" (1561), Blundeville mentions gallop, carrer, and gallop gagliardi which are all variations on the canter:
...to gallop strongly, to manege the single turns surely, and last of all to passe and carrer swiftly...to rein well and to bear his head steadily...learn to bound a loft with all four and to park with all, to gallop the gallop Gagliardi...
Markham (1614)
In "The Compleat Horseman" (1614 includes "Discourse on Horsemanship", 1593), Markham also highlights having lead legs:
In his galloping...His inward feet should play before his outward, and each of a side follow the other so directly that gallop may appear as the best grace of all his motions.