The key is that triggered abilities do not do anything at the moment they trigger:
603.2. Whenever a game event or game state matches a triggered ability’s trigger event, that ability automatically triggers. The ability doesn’t do anything at this point.
603.3. Once an ability has triggered, its controller puts it on the stack as an object that’s not a card the next time a player would receive priority.
Because of lifelink, you gained some life at the same time you lost a bunch. This caused Dina's ability to trigger, but it triggering doesn't do anything to stop you from losing; it would have to go on the stack and then wait until both players pass priority before it resolves and does anything. By this time, you have already lost the game due to being at 0 or fewer life.