I was trained to keep the lens horizontal, as you can get dust on the rear element by holding it lens-down, which can then be transferred to the sensor. This is also much more efficient when you're dealing with much larger PL lenses. One hand on top and bottom, handover, verbal confirmation and mounting. When mounting with this style, keep one hand under the lens, and then grasp it by the front to twist it into the mount.
Obviously, different styles for different teams, and as you work more and more with a certain DP or camera assistant, you will quickly learn their technique. For example, we typically leave the lens cap on until the lens is mounted to prevent any fingerprint smudges during handover, allowing us to do a visual inspection of the lens as we remove the cap.
Good video though, and I appreciate the emphasis on verbal confirmation, as it's something I wish more AC's knew to do.