OK Foot, bear with me here. The common name for the kick you have described is the, wait for it........................ "CUT KICK'.
Remember, I said there have been changes but a lot of times the old names still apply. I have, in the past, heard a lead leg side/round/front kick called a "cut kick". Whether using the heel or the ball of the foot, didn't matter. I have heard a lead leg side kick called a "push kick". That is why the USAT is trying to standardize the names of the kicks. Too many variations of the same kick with too many names, very confusing, neh?
If I tell one of my players to do a push kick and a round kick they will know the "push kick" is a lead leg front kick and the round kick is coming off the back leg because lead leg round kicks, whether static, skipping or retreating, have different names.
Confused yet?