It depends on who is doing the punch. For me we used our waist because we follow the concept that both arms are used to send one punch. For example. To send a right punch. The left arm should pull to help power the punch.
Another drill that I use is to focus only on the pulling and not the punching. The faster I pull my left arm back by twisting my waist, the faster my right arm goes out. The first time a person does the exercise, he/she will discover that the waist has completed the twist before the punch has started. This is often due to the lack of connection between the two sides of the body.
In application, I have the option to use that twisting motion to redirect an incoming punch or kick. I can also grab my opponent's arm and pull him into my punch. It is impossible to pull my opponent into my punch unless I twist my waist. If I only twist my hip then there is no pull. The pulling of one arm can power the punch of the other.
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Is it always like this? Of course not. Sometimes there's no need to pull that left side too far back. Most of the time the twist is short and powerful. I think I have some video of me punching the heavy bag using the long fist exercise that you showed.