It is not!
Miko receved the following from the Orlando Police Department's airport division a few minutes ago:
From: John O'GradyThus while you can't legally carry within the airport terminal, you can carry in the hotel itself. I just confirmed with the hotel that it can be accessed without entering the terminal.
Subject: Fwd: Airport Firearms Carry and attached hotel
Mr. Tempski, I am in receipt of the below email request. In short you are correct in your interpretation of the FSS. The Hyatt Hotel is it's own property so legally carrying a weapon is OK. Carrying a firearm in the terminal is prohibited. If you have any other questions or concerns please contact me directly.
Regards, John O'Grady Orlando Police Department Division Commander Orlando International Airport
Thanks again to Miko for getting a definitive answer from law enforcement. I should know not to trust public affairs and marketing!
UPDATE: Thanks to Markofafreeman in the comments below we have some directions on how to enter the hotel if you are driving without entering the terminal: If you drive, go up to the 4th floor parking, following the signs for the Hyatt. As advertised, the entrance is right there and you don't need to enter via the terminal.