What you write is the most common way to ask for a special gate. For an icelandic horse it is called tolt. For each gaited horse they have a different name for the same footprints, and they look a bit different because the size of the horse and the size of the legs.
Most icelandics have short legs and a tennesee walking horse have long legs, so it look completly different.
And the speed is faster or slower. An icelandic is the king in the gaits.
an example from a bitless ridden icelandic in the snow. I thought it is a good example because we have lots and lots snow over here.
I know this lady so she would not mind if i place her video over here http://www.youtube.com/watch?v=_FlywYT4QtI
When you shortening the reins watch out the horse is still using his hindlegs in the correct way, otherwise the back become hollow and that is not what you are looking for. It is quite difficult to explain this on paper.
It is a way of feeling how to aks for a special gait such as the tolt in a friendly maner.