On the last cruise I had "fun Play Credits" I had to go to the casino cage and claim them. Like you read, I also read that I needed to bring the e-mail or mailer that stated the offer, but I was never asked to present anything when I went to the casino cage, and they had my name on their list of cruisers that were due "Fun Play".
At the cage, they offered to either load the funds on my sign and sail card for use in the slots, or they would give me "fun play chips" for the table games. I took the chips, which are not the same as casino chips. The "fun play chips" cannot be exchanged for cash, and could only be used to play at the tables. I managed the use of the chips by always taking any winnings, which are paid out in standard casino chips, and only playing with them until they were gone.
That was my experience, but yours may vary.