Hello o/
Basically why you're getting that is because you're testing it in DEBUG mode via rpg maker (when you do it like that your character will be able to use all hms by itself).
Best way to test hm changes and most of the changes in the game is by playing the game without the debug mode.
Also you'll have to change the script in the beginning a bit so you won't get errors, should be something like these:
also you can add these 2 lines so you can also use the move from the pokemon tab
HiddenMoveHandlers::CanUseMove.copy(:CUT,:EMBER)
HiddenMoveHandlers::UseMove.copy(:CUT,:EMBER)
add them so it looks like this
also when testing make sure to add gym badge that is required for you to use cut (by default it is set to 1) or change the requirement to 0 in setting script section
jsut add a event with this script, and make sure to talk to it before trying cut :]
$Trainer.badges[1]=true
Have fun and good luck with your game o/