Mages: Types and balance

So I like mages. A lot. As I've played LoL in these last few years, I've noticed mages can generally be divided into four types:

1. Artillery mages. These guys will have high AP ratios, all or most of their abilities will be geared towards damage and they'll have lots of AoE damage. Good examples are Malzahar, Vel'koz, Brand and Xerath.

2. Assassin mages. These focus mostly on bursting really hard at fairly close range, usually against a single target. Most mages with good mobility are in this class. Good examples are Annie, Leblanc, Ahri and Kassadin.

3. Combat mages. These usually have to get up close and personal with their opponents, like assassin mages, yet their damage is less bursty. In exchange, they usually have decent CC, can AoE quite a bit, have relatively low cooldowns and can be tanky. Good examples are Cho'gath, Vladimir, Kennen and Mordekaiser.

4. Support mages. They tend to do less damage but can control or weaken enemies will protecting or strengthening allies. Good examples are Morgana, Lux, Lulu and Zyra.

With a few exceptions, all mages fit neatly into one of these categories. So, what's the point of all this? Well, the mage class itself is ill defined, usually only described as "uses abilities a lot". However, a mage's role within a team will change a lot depending on what kind of mage they are, so it becomes important to be able to tell them apart. It's also important to be able to tell their role when balancing them, or creating new champions.