If your skin is peeling off then it is most likely that you are swiping your punch. This means your impact isn't going directly into and directly out of the punch. If your punch slide across the surface then it's like striking a match and your skin comes off. Everything must go into the bad and directly off it.
I am referring to just normal jab and reverse punch here. I know about going directly in and out, aiming a few inches into the heavy bag and focus the force to that point and get maximum penetration. I have been practicing this for years. That a good punch produce max penetration with very little movement of the bag going backward. This is basic stuffs.
BUT, you don't punch a single punch and stop the bag from moving before the second punch. I practice 4 or 5 punches in a sequence like jab, reverse punch to the face, then 3rd and 4th to the stomach and the 5th to the face. The bag is going to move particular when punching the stomach, one punch to the left side and one to the right. The bag is going to swing.
When the bag starts swinging some, you NEVER get 100% in and out, so you get the sliding surface even if you hit a perfect straight punch. If the skin on the knuckle is not trained, you are going to break the skin sooner or later. This not only happen on the first time punching, If you stop punching with bare hand for a month or two, you WILL break your skin if you do it hard for a few rounds on heavy bags. This is just the nature of the game. Just look at this video you linked. Look at how much the bag moves if you punch multiple punches:
If you don't want to break your skin, maybe putting a small bag on the wall and punch one punch at a time. That won't move and you are in control how straight in and out of your punch. But what is the use? The important thing is to punch a series of punches in the attack and you can focus to punch on the spot even if the bag move some.
Lastly and more importantly, I am 5'5", I have to learn to punch up. My punch is never perfectly true to the surface to the bag. I have to prepare to fight someone that is 6' or taller!!! So, by default, it's going to be more abrasive to the skin. In this note, I don't understand people punching the bag at the neck level.
It's so so much easier to punch a good looking straight punch to the neck level. Look at the video, a lot of his punches are to the neck, not to the face level even for his height. But still, when the bag moves, it's going to burn the skin. For me, I have to punch up. Believe me, it's a lot harder to punch up.
Back to the circle punch, when you hit the bag that is moving ( like in the video), the bag is going to rub the skin even if you circle punch perfectly.
You taller people take for granted and might not know how hard to punch up. Try it and you'll see. For shorter people like me, If I punch to my neck level, that's punching the upper chest of a tall guy!!! For what?