How to jump nose up while on your back

In your example video, he is forward-flipping while upside down. The reason he doesn't flip 360-degrees like usual is that he flip-cancels by immediately holding the opposite direction (down) on the joystick.

[Edit] I made an example video including controller inputs so you can see how it's done.

It's actually more difficult than I expected, because if you try to forward-flip while your entire roof is touching the ground, it won't work. To work around this, you boost downwards into the ground at an angle so that the front of your car bounces up just before your flip. The example video makes it look so effortless...

[Edit 2] Actually, looking again closer, what he does is slightly different. He boosts into the ground, then foward-flips before he hits the ground, and flip-cancels so his car drags along the ground during the flip. Then, before the flip is complete, he un-cancels the flip (by letting go of the joystick) which causes his car to rotate for the remainder of the flip. It's a tricky move which takes a lot of practice, because it requires precise movement and timing to pull off.