Well it's not horrible
but there are certainly things that can be worked on.
First off, I'm glad you've already taken him back down to a trot, because that is where he needs to be right now.
Overall, I would also try to keep your hands off him a bit more, and instead, use your legs and body aids. And/Or work on softening him up a bit. I see his head up in the air a lot and his back hollowed out. He'll turn and perform better if you can get him collected up and more rounded in the body.
On your first barrel on the first run, you need to finish
your turns. He drifted out on the end of the turn. Your pocket going into it seems fine, so just make sure he finishes. A drill you can do to help with this is circle the barrels twice (or more) so that he doesn't rush off to the next barrel, and he learns to finish his turn completely. Also, make sure you use that outside leg to push him in toward the barrel if he's drifting.
On the second barrel of the first run, I would really like to see a solid, distinct rate cue. Most of what I saw what more pulling on his face, his head goes up, and he bounces roughly into the turn. In between the barrels, you want your body forward, and encouraging him. Right before you get to your pocket, you want to obviously sit your weight into your seat first. If then you need to check him a bit with the reins, you can, but as I mentioned above, working on softening his face and getting his body rounded will help alot.
Second barrel this time you finished!! A wee bit too close, as you bumped it, so really work on getting him cued into your legs, so you can push him closer or farther ffrom the barrel as you need. You can work on spirals (away from the barrel pattern) to use your legs to teach him to make the circle smaller or wider when you ask.
And the gallop home from the 3rd barrel, you see how high his head, how much contact you have on the reins, and hollowed out his back is? Some general training (away from the barrels) can help improve these things.
Second run, first barrel, was a bit smoother. I would still tighten up the backside of the barrel just a tad, so you are leaving the barrel just a tad closer.
Second barrel second run, I would actually make your entrace to the pocket a bit closer to the barrel. You are going in just a bit too wide, which is causing you to come out too close at the end of the turn. You also tipped his nose to the outside just a tad before you made your turn. That's not helpingn the shouldering in problem. And this is where making him responsive to your leg will help. Instead, you want to do more of a two-track manevar if the horse is turning too soon. Tip his nose to the inside, and put pressure on his belly with your inside leg. This creates the correct bend in his body for the turn, and your inside leg and move him away from the barrel.
So mostly, work on making a perfect pattern every time at the trot. Also, make sure you do the same body language as you would as if you are galloping. So even if you are just walking the pattern, lean forward as if you were running. And make sure you make your rate cue obvious. Sit your weight back, check/collect with the reins if you need, and just do the exact same thing every single time at a slow pace. So when you speed it up, those cues will be the same, your horse will know them, and will responnd accordingly.