The Turboglide
was troublesome. Buick called their version the Flight Pitch. Their were several transmissions all called Hydramatic's, as it was GM's division which made them all.
A few could even be push started, because they had a front, and a rear pump. Those were history by the late 1960's.
The 2 speed Powerglide was the most reliable and efficient before the 3 speed Turbohydramatic 350, and 400's debuted.
There was one called the Slim Jim, one called Roto-hydramatic used in the '60-'63 B.O.P. compact cars, a Super Roto-hydramatic, often confused as a Slim Jim, as both had an emptying fluid sump from first to second gear.
Check out Charlie Tranny site. He explains them all well enough, and has some nos/rebuild parts as well.