Throughout my long playing career (if you can call it that), I have been fed any number of what now seem like crackpot breathing directives, including "Ya gotta push hard out your butt, like you're tryin to fart." "Draw in that sphincter muscle and make it real tight." "Support has to come from the diaphragm, don't worry about your chest."
I've finally begun to understand a couple of fundamental things: 1) Both your chest (lungs) and diaphragm have to be fully engaged and (2) It's not the volume of air that opens up the upper register, it's the velocity. Plenty of air has to be available and you have to be able to generate great airflow speed. Your mouth cavity and tongue also effect the rate of speed.
We get very obsessed about our chops, but in fact our chops don't kick in until all the above happens. It takes a lot of strength to resist a small, concentrated, fast-moving column of air. The job of your face muscles is to allow your lips to either tighten or relax, to produce faster or slower vibrations that are then amplified by the trumpet and emerge as notes of different pitch.
Whatever system of playing works for you is probably the one that allows this system to operate with the greatest efficiency for your particular physiological and psychological makeup.
Here's another video, just to give our friend the diaphragm its due. Ten points to whoever tells me where that extra "g" came from:
So my friends, breathe deeply and prosper.