I am an oldtimer and grew up riding a "singlefooting Morgan". Morgans were held in high regard where I come from.
The term "singlefoot" does indeed mean that only one hoof is on the ground while the other three are off the ground.
Singlefooting was (and still is) associated with gaited Morgans - which have been around forever since General Stonewall Jackson rode one in the Civil War. Gaited Morgans
The above link does use the term "singlefoot".
Sorry to burst the bubble of the folks that think Morgans just trot. People have been trying for decades to breed the gait out of the Morgans and also the "Indian Shuffle" that certain Appaloosas perform. Thankfully they have not succeeded
Singlefooting is not considered racking, it is not considered "what the horse must be doing if it isn't performing a runningwalk or foxtrot, etc".
It is a legitimate gait that's been around forever.