You're correct that the PLAY AGAIN button is part of the UI.
In sweepstakes events you just have to enter the tournament and complete one round. That satisfies the entry requirement. It doesn't matter what your score is as long as you have a score. It doesn't matter how many subsequent times you play after that.
If you choose to play again you aren't submitting another entry, you're just playing again. We leave the sweeps unlimited so players have more options for play and more leaders to chase down. You can only physically enter each tournament one time (when you agree to the terms and conditions of the tournament).
What that particular passage is alluding to is someone that is entering the tournament using multiple accounts. That would be considered more than one entry and we will disqualify the winner if we discover they have entered more than once.
Play on as many times as you like with no worries. =)