Cut is a Hidden Machine while Slash is not. Slash is a more offensive move while Cut is more a move for HM slaves. Many moves could be usable overworld, like Earthquake destroying the Victory road, free fall as a replacement for Fly but overall this is not possible. Moves that are usable outside of battle are very limited. Also Cut and Slash have different properties, Slash being one of those attacks with 70 base power, 100 accuracy and higher critical hit chance (the other ones with these properties being Night Slash, Psycho Cut, Cross Poison and Shadow Claw) while Cut only has 50 base power and 95 accuracy. Also, Cut is named "Cut" and Slash has another definition.
According to your ideas, Bodyslam is also just a stronger version of Tackle, Earthquake of intensity, Rock Slide of Rock Throw, which are all possible with common sense but all have different definitions.