The $2 was taken at the gate where the taxi's drop you off. Each beach chair or umbrella is $10 each. We paid $30 for 2 chairs and 1 umbrella.
There was a lady who came in the same time we did who was just there in August and said they were not charging this fee then. The taxi drivers will also tell you this and encourage you to go to one of the "free" beaches.
We did not eat there or use the pool. It got pretty crowded by noon, but there where 6 cruise ships in port that day.
I personally was not thrilled with Royal Palms. It is a beautiful beach but it was crowded and loud. Jet skis coming in and out and just a lot of people. We had a better time at Chankanaab in Cozumel.
A couple riding the tender back with us went to Governor's Beach and loved it. They said it was quiet and not many people. I wish we had went there.