First off, there are no official terms for most things to do with Chao. I'll list the commonly understood fan terms, and also terms that are used on this site.
Blitz Chao: A fan term for the Duotone White when evolved to Dark.
Chameleon Chao: An old name for the Pure Normal.
Chaos Chao: One of the three immortal evolutions found in-game.
Duotone: A multi-coloured Chao. Sometimes called a two-tone.
Egg Chao: A Chao that is an entire egg, seen in Challenge Race. It is only available by hacking. Reincarnating a masked Chao will not make him an Egg Chao.
God Chao: A myth, it does not exist.
Guardian Chao: A fan term for the Duotone White when evolved to Neutral.
Liberty Chao: A myth, it does not exist.
Item Trick: A means of using an animal or Chaos Drive multiple times.
Jewel: Can refer to either Shiny or Metallic Chao, but Metallic is correct.
Metallic: A Chao with chrome-texture skin. Sometimes called a Jewel Chao.
MTN Chao: MonoTone Normal. A Pure Normal Chao.
Neutral: Chao alignment midway between Hero and Dark.
NiGHTS Chao: Commonly used term for the basic Neutral Fly evolution.
Official Chao: The limited edition Amy Chao, Tails Chao and Knuckles Chao. Tails Chao can be got in Phantasy Star Online, and the other two are only available now by hacking. Not to be confused with Sonic, Shadow or True NiGHTS. Like Chaos Chao, they are immortal.
Pure: A Chao that is one flat colour. Sometimes called Monotone.
Pure Normal: A regular Chao with the Pure gene. Sometimes called Chameleon Chao or MTN Chao.
S-rank Chao: A Chao that has a grade of S in every stat.
Shadow Chao: The Dark Run/Run evolution. Some people do not like it being referred to as Sonic Chao, since it is not an official Chao.
Shiny: A Chao with a sparkling skin. Sometimes called a Jewel Chao.
Sonic Chao: The Neutral Run/Run evolution. Some people do not like it being referred to as Sonic Chao, since it is not an official Chao.
True NiGHTS: The Neutral Fly/Fly evolution.