Magic Damage vs Physical Damage
The advantage to having two types of damage is that you can have two different types of attacks/champions. Someone that is able to conjure magic or really anything outside of something that is physically possible would be magic damage and stabbing or shooting someone would be physical damage. Why is this not consistent in League? If the answer is balancing, numbers can be tweaked. In League, you should be able to see an ability and know right away whether it's magic or physical damage.
For example, Darius's Q spins his axe around. That does (and should) deal physical damage.
Fiora's Lunge does (and should) deal physical damage since he is stabbing you with her sword.
Brand's abilities all deal magic damage, which makes sense. He is creating fire that he is throwing at you. He uses magic.
These are just a few what I feel should be changed: Fiora's W is magic damage but she is stabbing you with a sword. Katarina's Q,W, and R should deal physical damage. I can understand the E doing magic damage since she is "magically" appearing behind the enemy.
Other notes: Anything that is not a physical weapon can deal magic damage (such as poison). I'm okay with poison doing magic damage instead of physical damage. I don't care about AP or AD ratios. For example, on Katarina, you can keep AP ratios and just make it deal physical damage.
The point is that you should be able to see an ability and know instantly whether it's physical or magic damage.