Well...I think that is probably your problem. Some hard core guys make fun of mirrors in a Gym, but they serve a purpose at times. Look straight into the mirror, not up at the bar. This will keep your neck straight and also stop you from leaning backward.
If you have no mirror you can do the same thing by picking a spot on the wall directly in front of you and avoid looking at the bar.
I'm not a fan of TUT for barbell presses. I think you need to move that weight at a good speed and not worry about counting seconds
I bet this solves your problem!