Caillois's Terminology

Types Of Play

Paidia

Controlled by improvisation and spontaneous ideas.

Ludus

Controlled by structure and rules.

Forms Of Play

Alea

Based around chance and "Luck", e.g. Slot machines.

Ilinx

Based around Vertigo, things that alter our perception, e.g. Taking hallucinogens

Agon

Based on competition, e.g. Chess

Mimicry

Based on roleplaying, e.g. D&D


Overall this terminology is somewhat useful, but to use it too frequently would only baffle and annoy those who have not learnt it, it has its place, but its place is not common, on that note I also had a very hard time understanding "Ilinx" or thinking of a game it applies to.