A 2/4 shimmy as I learned it goes up & up higher on the right hip, back to neutral, then up and up higher on the left, back to neutral. I think you can also do a 2/4 shimmy down. The count is similar to a 3/4 shimmy as we do it in ATS, but you don't return to neutral in between hip lifts on one side, just when you switch to the other hip. The 2up 3down step reminds me of this shimmy, at least the first part.

A more cabaret way of teaching 3/4 shimmy is as up, down, up (vs the ATS up, neutral, up, so you pass through neutral to an actually dropped hip), so if you'd learned 3/4 and 2/4 this way, the ATS walking shimmy would seem like somewhere in between the 2.

Concur. 3/4 absolutely.

I call the FCBD 3/4 shimmy my ATS 3/4, as I've learned 3/4 shimmies in several different ways and dance in an area where there is some BD genre crossing so it helps me to keep them all straight. When I took a workshop with Mira Betz last summer she explained 3/4 shimmies in a completely new way that I'm still trying to wrap my brain around.