I'm pretty sure calling it a kipping press and performing it in front of a big mural that says "The Mecca of Bodybuilding" with lots of associated pictures won't make Rip particularly proud.
But the press itself isn't bad. A little more reach with your hips is in order, and timing the press with the bounce slightly better - you press just a split second after you bounce off the hip-rebound, so you don't fully utilize the energy of the hips. You timing is pretty good, but you could catch more hip-help if it were just that little bit better. Try doing a few reps during your empty bar warmup, and maybe the first one or two lighter warm-up sets where you just reach and bounce, without actually pressing, and feel the bar want to move upwards. Feel that movement, and then press right at that moment.
Also think "aim for your nose" as you press the bar. You press just slightly away from you off your shoulders and then have to move the bar backwards again as it goes up.