I love how he runs with his tail up, just like my mare! Go Arabs! Lol
Your horse runs ncie & normal, though, whereas my mare Chico runs like Peppy Lephew (or so several people at my gymkhana group as well as my dad have been kind enough to point out!)
As for his gallop, in a hand gallop the horse should not pick up enough speed to lose the 3-beat rhythm of the canter. Your horse seems to be hand galloping, although in the 2nd video, near the end he picks up some speed ;)
Here's a video of a hand gallop to compare.