A lot of different places do uke differently, and for different reasons. One isn't necessarily better or worse than the other (though it can be).
I know you didn't say you wanted comments about your teacher, and I'm not making a qualitative statement about his technique, but I will say many of those techniques aren't the easiest or prettiest to fall out of. Looks like you guys were having a semi-informal "saturday afternoon' type of class and were just riffing a little. So if you felt a little stumbly and out of sorts, its probably because you were getting put into awkward positions at awkward speeds. Its a lot easier to fall out of something where you are coming in full force, and Nage is sending you on your way, or stoutly redirecting you.
You are dropping to the mat really fast when released. I'm not sure if that's the way you are taught ukemi. If so, fine. If not, try to extending out a little bit as you are getting sent on your way. This will help you get into better form and help you meet the mat at a better angle.
I'll second what was said about the slapping hand. It looks like you are attacking the mat. A lot of people do this, and I'm not sure why exactly. The point of that hand is to distribute momentum, and slapping it barely does. Think like pressure point strikes…hit and stick. The purpose of that is to distribute energy from your body to theirs. Same deal for ukemi's slapping hand - distribute that momentum through your arm to the mat. If you can get your arm hitting the mat well before your body, have it stick, then let your body come down a slight angle from your arm, you will be landing a lot more controlled.
Edit: Most important is to train how your teachers tell you. If you have a sensei or senpai willing to sit with you and do some video study, that would be best.
This isn't the best example, as its almost all jumping break falls, but it shows the idea of getting that 'slap hand' out well before the body and landing softly.
Another really important note, though not necessarily directly relevant to the video you showed, is to stick with your Nage as long as possible. This allows him or her to execute the full experience of the technique. If you fall too early, and collapse really easily, it takes away from both of your training. Now when you are doing technique casually, it can be confusing as to when the thrower is releasing or transitioning to something else. But when your teacher starts throwing you with a little more energy you will definitely know where you are supposed to be going, lol.