Air kick is a definitive ‘kick’ from the abdomen like yelling “Hey”.
My take is Air Support is more ‘following’ the Air out to maintain compression. If you can be full of air then ‘Air Release’ a squeak a note from your lung compression alone, try adding ‘Air Support’ afterwards to increase volume. This idea is throughout Greg’s exercises. The way we all learnt as beginners used to be blow more or ‘kick’ the air to get higher. This obviously can work to certain level however, if your aperture corners are not strong enough, you will need mouthpiece pressure to get the higher note, which we all know is limited.