You are almost correct.
But the machine reads/writes to the card, so just cloning the UID isn’t enough.
What the staff does to reset a used card, I have no idea. They probably have a programming unit in the station, so they can program the cards based on the customers wishes. ‘do you want the, “triple platinum super duper wash” or “some other fancy name here” wash?’
So in theory, the process is:
1: buy a 5 wash card
2: crack and clone it, and save the dump file
3: gift the original card to someone you like, as you don’t need it anymore
4: use your cloned card for four washes (the machine tells you the number of washes you have left each time you use it, so when it says you have one wash left:
5: Re-upload your dump to your card, so you have 5 washes again
6. Repeat steps 4-6 every 4th wash