What do 
all have in common?
According to Riot they are all classified as mages.
So that makes me ask, what in the blazes is a mage really? I'm pretty sure Janna isn't a mage like Ahri or Brand. Same could be said for Sona (just poke) or Soraka (riot just nerfed her mageness and told solo lane Sorakas you're still supports).
What is a mage really? Why is it that Ahri, Brand, and Cass can do damage but the ones in the last part of the list can't?
All of those champs have been AP mids at some point with the exception of amumu and nami. Also they all use magiks.
They mainly do magic damage from spells. Mages generally build AP
You could hover over "Mage" role in team builder and it will give u more of Riot's definition of a Mage and what they do.
Generally, a mage is a champion who primarily works with their abilities. Their abilities often have utility as well as raw damage.
A marksman primarily uses basic attacks, a fighter uses a mix of basic attacks and abilities, a tank is primarily concerned with being hard to kill, a support has abilities that help their teammates, and an assassin is someone who does a lot of fast, often single-target burst damage.
All champions have two roles, so Katarina is an Assassin-Mage, while Amumu is a Tank-Mage.
Well this is my opinion only on the topic, but kind of use your imagination. A MAGE, usually means that they use their "Mysterious Power" to do things, rather it be for damage, utility, or even healing. Usually a champion with a Mage Tag scales the best with Ability Power and is almost useless when building attack damage. When I think of ASSASSIN i think of two things, they are casters, but their "Mysterious Power" deals more with things you would imagine to be super human. For Example, 
Talon's "E" attack is implying that he's just THAT FAST (Or stealthy).
To me, the only confusion as to what a mage is is when you compare them to assassins, so I'll try to explain both:
An assassin deals a lot of damage, usually single target. These champs focus on closing the distance between themselves and a high-priority or out-of-position enemy, and has to combo all his abilities and attacks to take him from 100% to 0% as fast as possible, bursting him down in the blink of an eye. They are usually equipped with a gap-closer, high single target damage, and an escape.
A mage, on the other hand, deals all of his burst from afar, capable of dealing high AoE damage to the entire enemy team. Maybe an assassin's full rotation has more damage than a mage's rotation, but the mage has it up sooner and can dish it out more safely.
