Can't get the Cut HM in Pokemon Green

Solution 1:

To get the Cut HM, once finding your way to the top of the ship, and defeating your rival, the captain will say he's not feeling well, and ask you to rub his back. Make sure you exhaust his dialogue (you may be presented with a "Yes/No" choice, I'm not too sure. If you are, choose "Yes"). You will make him feel better, and he should reward you with the Cut HM (HM01), regardless.

The only restriction might be that you will need to beat Misty (Cerulean City Gym Leader) in order to obtain it, but I don't believe it's necessary. Defeating her allow you to use the HM outside of battle.