>>50925707JP name of Cut is "Iai Giri", so that would be the kind of theming to expect.
Assuming it's received early on, you could use it to teach mechanics to the player, like how critical hits work. Maybe an auto-crit, but only if you go before the opponent?
Weirdly enough, the JP name of Hyper Cutter ("Kairiki Basami") has nothing to do with Cut, but is related to Strength ("Kairiki"). Also Machamp's JP name is KAIRIKII.