Your form is not horrible. Fear not. You have decent timing. You try to move your hips, then drive the bar up. Unfortunately, you don't really move your hips. You just overextend your spine. Look up the press 2.0 video on this website. It will be helpful. Here's what you need to do.
- Start with a neutral spine. You are arching your back a little too much at the start.
- Lock your abs down hard.
- Reach forward with your hips. More than you are now. Don't lean back and crank the hell out of your spine. Make your abs do their job.
- Aim for your nose with the bar.