When I started in SKK, I too was taught the c-step when punching. Pretty much every technique was stressed with this method, and I never saw anything done with the punch being thrown as a cross.
When changing to the Parker system, the techs. were described as step thru punches, not c-steps. When I'm throwing a punch for a tech. I don't c step, I step.
Interestingly enough, take a look at these 2 clips of Short Form 1 from the Parker system. In the first clip, we see steps, when transitioning from move to move, while in the second clip, we see the c-step.
When changing to the Parker system, the techs. were described as step thru punches, not c-steps. When I'm throwing a punch for a tech. I don't c step, I step.
Interestingly enough, take a look at these 2 clips of Short Form 1 from the Parker system. In the first clip, we see steps, when transitioning from move to move, while in the second clip, we see the c-step.