I did notice, however, when you're riding at a trot there at 0:38, from that point on you really bump your horse's sides with your legs. Notice the man riding with you. Watch his legs. See how tightly clamped they are to the horse's sides? When you ride, you should ride with a constant and steady pressure with your legs, and steer with your horse's ribs more than his face. I noticed too, every time you move your hands your horse reacts to the bit. (Which should be, naturally, the case, but when you're riding pleasure, you should be able to ride no reins at all) Lift for go, lift for whoa.
Watch your feet at 2:49.
If you really focus on driving him with your feet, I think he may come along quite a bit more for you, he won't be so busy thinking about why the bit in his mouth is wiggling a little. And it may keep him from being so inconsistent with his carriage.
Also, I'd give him a little more rein, especially at the canter, I noticed there at the end of the vid he was tossing his head a little at the contact with the bit.