Just to give you an update on the situation. They gave me a really hard time upon boarding the cruise ship since I‘m not a US citizen and they couldn‘t find the entry stamp. At first they weren‘t going to allow me onboard but then a second supervisor came and the Esta confirmation was enough to board. If I hadn‘t had a printout of my Esta, I don‘t know if they were going to let me onboard.
So, for future reference, non-US citizens, get a stamp and have an Esta printout if you‘re doing a cruise.