A saut, as defined in Orchesography, is a jump, in which both feet are elevated from the ground at the same time. The definition emphasizes the position with elevated feet, which seems to imply that when the petit saut is referred to in step descriptions what is being described is mainly the moment of elevation, and does not necessarily refer to the landing or to the rising action that causes the saut.
Jumps are often categorized by their size in dance instructions.
This describes a step that is accomplished via a small jump. Arbeau gives the example of moving from pieds joints (feet flat, heels together) to grève droite (kick with the right). You could shift your weight to the left foot and then kick, but it is more lively (and arguably easier to balance) to rise momentarily from the left foot, repositioning it on the floor as you do the kick with the right foot; that is a very small jump. The petit saut is assumed by default to occur when you make such a step with one foot moving in the air, and is usually not included in dance tabulations
The saut majeur is a large jump, with a much higher elevation than the petit saut. This kind of jump is more visible, takes more time, and is notated as a dance step in its own right, for example in the galliard.
The saut moyen is a jump of moderate size, regarded as similar to a saut majeur but smaller. It is done in the tourdion