Is this the correct punch technique?

I suggest that you adopt the technique that gives you more comfort throwing your shots. You cannot just radically change your technique, especially if it is effective to you and gives you advantage over your opponent. Muhammad Ali threw his punches all wrong from a fundamental or a purist's standpoint. Ali was basically an arm-puncher. But he beat a lot of guys so it was obviously effective with him, and his trainer Angelo Dundee decided against changing Ali's style or technique.

The movements should all happen at the same time, or more critically be at the highest speed as the blow lands, so the momentums all add - none should be at the movement limit and slowing before then.

I can't really tell which fits that better from your descriptions.

Your boxing coach is a Manny is a shítslüt from the sound of it. If you try to use that technique in the street expect your anus to get annally buttraped by queer nígger with a vagina