Our first cruise I was basically clueless about, well, everything. I knew there was a formal night, but didn't understand the logistics. I (wrongly) assumed that you dressed formally on the night you ate at the more formal restaurant (Tritons or Lumiere's - this was before the Dream class). Imagine our surprise when we showed up at Parrot Cay in our casual clothes when everyone else was dressed to the nines!!!
To their credit, our servers and table mates were very kind. We all got a good laugh about it and had a lovely dinner. We did dress in our formal attire when we ate at Tritons, while everyone else was casual and we all got another laugh.
Live and learn! OP, your question has already been answered, but kudos to you for seeking clarification instead of just bumbling though as we did!!

I understand now why it's a ship wide formal night regardless of where you dine, but at first it seemed odd to dress so formally in a casual setting.