If you were using Ring of Destruction according to the text it had before its power-level errata (which this program does have, labeled as an anime card), you would be correct. However, with the new text, the resolution you saw is correct. The current version of Ring of Destruction does not damage both players at the same time. It damages its user first, then uses the damage it did to the user (if any) as a reference to burn the opponent for the same amount. So in this case, when Barrel Behind the Door is chained to Ring of Destruction, during its resolution, Ring of Destruction attempts to burn its controller for 3000, but Barrel Behind the Door redirects that damage to the opponent. Ring of Destruction then sees that it didn't do any damage to its controller, so it does not do any more damage to the opponent.