Thank you alexb133 for the information.

There are chances that in the future I move to a H1B visa, so it's good to know that if I want to visit Cuba legally I have to do it before changing my J1 visa.

Even if Cuba doesn't stamp my passport Cayman Islands will. I think they do both entry and exit. So it will we suspicious if I only list the Cayman Islands and I have an entry stamp, then an exit stamp the day after, then an entry stamp the week after, and then an exit stamp the day after.

If the only thing that is going to happen is that I will spend some time at the secondary inspection, and that is 100% the ONLY thing that will happen to me, I might not risk my situation by lying on the customs form. I booked flights with a two and half hour layover and I will be traveling just with a backpack...

Also, everything I buy in Cuba (basically souvenirs, not cigars) I will send to my country in Europe from the Cayman Islands. And this might seem too paranoid, but I will bring an extra SD card to copy all the pictures I take in Cuba and I will send it along with the souvenirs. I will bring no laptop with me.