i actually only just realized that witch hats and wizard hats are the same hat. Different color maybe
I think wizard hats are more optimized for the peak, to the point where it can still count as a wizard hat if it's all peak, whereas witch hats are more optimized for the brim, and it can still scan as a witch hat even if the point is very modest as long as it has a wide brim. so it's more like they're the same species of hat but display some modest sexual dimorphisms