The dragon is Draco, and the book is Heaven Cent by Piers Anthony, part of the Xanth series.
You repeatedly hear characters saying things like 'As sure as Water Douses Fire', only to hear another say 'As sure as Fire evaporates water' -- eventually, Dolph makes them realize that they had a miscommunication, and the dragon chooses to give the widow of the Merman the pair of valuable opals that was behind the game.
Some relevant segments:
"Mela thinks you just attacked him and stole the opal," Dolph said. "She thinks you're a rogue dragon."
"There were no witnesses to the event, so I suppose she can think what she pleases. But I have told it as it happened. I threw fire, he threw water, plain as two days. I am absolutely sure of that, as I am of his bad attitude. That gem is mine!"
But something was nibbling at Dolph's memory. "Mela— she said something—I think it was just a saying she used— about fire and water. 'Sure as water douses fire'—something like that. I wonder—"
"That's backwards!" Draco growled. "I told you, fire evaporates water. Every time. You breathe fire on it and it heats and turns into steam and it's gone. When I come home and drip on the nest, I dry it out by heating it a little."
"But maybe to a merperson, who lives in water, it would seem the other way," Dolph said. "I remember something else she said, about sand displacing water—"
"That's backwards, too! Water covers sand! I'd heard that merfolk were addle-brained, and this proves it!"
Then, a page or so later:
"But what I was trying to remember—Ah, now I have it! Fire melts sand, she said. So you know, they—"
"All backwards!" Draco exclaimed. "What idiots! They think fire melts sand, sand displaces water, and water douses—" He broke off. "Uh-oh."
"I wonder whether they play mat game the same way?" Dolph asked innocently. "Backwards?"
"Suddenly I very much fear they do! So that when I threw fire and Merwin threw water—"
"You mean he really did think he won?"
Draco snorted out a fierce jet of fire. "Oh, my," he hissed. "I wish I'd understood! We could have discussed the rules, gotten them straight . . ." He trailed off, his words fading into thoughts.