This video explains how the crafting system works, including the corner cases that are impossible to infer from the in-game text.

11/20/2017 UPDATE: Since I posted this video, the game was changed so that Asharra's Fire Shield DOES get the damage boost from secondary weapons. This was never mentioned in any patch notes that I saw, so it was a stealth change (unless I missed something, in which case, my apologies). This makes the bird mage's best ability even better!