the taxi fare is actually around 18 AED....
from what I remember from last time. I know I had to pay 40 AED coming FROM the airport since all airport taxis start at that price. On the way back its just a normal metered cab ride and from the JWM around the corner it was definitely < 20.
In any case, you did handle the situation well. My guess is the doorman or bellboy or whotever gets a small cut from the private cabbie.