I can't comment on the video, but is your horse naturally lazy or when you are walking him on the lead does he walk faster than you and you have to constantly check him back?
My mare is very forward. When I practice, I use a rope halter and a 12ft. Lead. For my mare I would walk, then walk fast and then get my life up (body language) and then I would kinda jog in place (about the speed that you were walking) then I would "click to her" and begin moving more forward-out front. We had a couple times of me going forward at the jog and her not picking it up, so there was a hard pull on lead and me coming to a stop pretty quickly. Then I would do it again and by the second time or so she was picking up the trot. Now when I want her to go from the stop to the trot or from the walk to the trot I only have to move my arms slightly forward, maybe a quarter of an inch, no clicking and we are off trotting. Of course we have been working on this for about 9 months, it didn't come over night. I work my showmanship every time I go to ride. So we are practicing some part of it at least 4 times a week. Either perfecting the trot off, or our turn on the haunches or whatever I feel we are weakest at.
She is still a little lazy turning on her haunches. So, I get her moving, sometimes I need to give her shoulder a tap and then we go back to being in frame and moving around. I always watch her left hind to make sure she does not move it, if it looks like she is going to move it, I have my que word "here" to tell her not to do it.
As for the head up, my mare used to do this, because it was the only way to get her to stay set back on her haunches to pivot. But as she has gotten better at the pivot, I have lowered my hand and made her turn with her head lower. Again, I really watch that back foot and if it looks like she is going to move it, I say "here" and stop her before she does then we continue on. Of course this is in the practice, once we are in the show ring there is no stopping, just praying she doesn't move it. Sometimes you need to feel whether they need to be asked forward a little bit or do they need to be pushed back a little bit.
I also always set her up when I come to a stop. Whether she is in her halter or I have dismounted from riding and am on my way back to the hitching post. I ask her to set up, this way she knows that when we come to a stop she needs to have all four feet lined up. She now sets herself up about 70% of the time, I don't have to ask her to do it. Hopefully by the time show season really gets going this year, she will be at 95%.
As for a big halter, I was lucky enough to find mine used at a consignment store, it was still $150. But if you go to Stateline tack or some other online tack store, if you just buy a nice leather halter, you should be fine. Especially if this is your first year showing and it is just going to be around your local area. The judge really doesn't care how much silver your horses halter has on it. They want your horse to be clipped, bridle path, face, ears (at least around the outside, unless they are really, really fuzzy), and around the coronet band. Unless they have white socks, then shave those up as well. Also is your horse shiny and not necessarily Show Sheen shiny, healthy shiny, tail clean and mane laying down and looking nice. Also the halter needs to fit properly, this may mean putting in some additional holes over the pole. The cheek strap should fit snuggly under the jaw, the nose band should be up and fitted closley to the face, not hanging down. Tried to find a picture to show you, but couldn't locate any. I'll look some more. Good luck.