The first video the reason why it is raising up like that is because you have the switch set to Auto, you cannot be in No Cut and Auto at the same time, would need to be set to Manual. It looks to be working fine when set to no cut, torch is moving up and down like it should, the next video of it cutting the torch does not raise up between pierces. It would seem that the off signal is getting interrupted by interference, make sure no other cable crosses over the machine torch cable especially the height control motor cable, keep these separated. Also try the cut in Manual Mode and see if you get better results.
With a new 45XP double check you ran the arc voltage wires to the inside of the Hypertherm, cut in Manual Mode with the switch set to Read watch the arc voltage display, switch the cut to Auto and Set, adjust the voltage to what it was reading in Manual, then back to Read. This is how you would need to operate the red screen height control before cutting in Manual.