What a gait actually specifies
A gait is the repeating pattern of when each leg lifts, swings through the air, and plants back down during locomotion. It's a timing and coordination schedule, which legs are in stance phase, touching the ground and bearing weight, and which are in swing phase, moving through the air toward their next placement, and how those phases are staggered across all the legs.
Different gaits are really different answers to the same underlying tradeoff, how much of the time is spent with a wide, stable support polygon versus a narrow or nonexistent one. A slow, cautious gait keeps more feet on the ground simultaneously, maximizing stability margin at the cost of speed. A faster gait reduces how many feet are down at once, and some gaits, like running or a quadruped's trot at speed, include phases where zero feet touch the ground at all.