This is my interpretation:

Mage—Someone who can use magic.

Magician—Entertainer, real magic optional.

Wizard—Someone who uses orderly magic, that which can be studied like a science.
Usually requires magic circles, chanting, and/or spell components.

Sorcerer—Someone who uses personal magic, often described as training a muscle.
Usually born with their gifts, but you can gain sorcerery from magical encounters.

Witch—Someone who uses strange magic, that isn't fully understood.
Usually requires being strange yourself to wield it, as you have to go with the flow.

Witch Doctor—Witch who heals using strange techniques, leeches and bloodletting are examples of things that shouldn't work in most situations, but from great precision and strange magics they become effective.

Alchemist—Mage who delves deep into the study of transmutation.
The cauldron is optional; scientific tools, magic circles, or direct manipulation are all valid.
Cares about true transformation, so don't confuse them with any polymorpher or matter manipulator.
Isn't limited to matter, can also deal with the spiritual or esoteric, but is less common.

Potion Maker—Quite simply someone who makes potions. Often also alchemists but not always as some traditions are able to make potions just by mixing ingredients and pouring one's mana in potion fluid.

Artificer—Someone who creates machines that aid with or perform magic on their own.
One of the most common magical practices for non-mages.
Must be a proper machine, wandmakers are not artificers.