To me...
Stuff from your Grand Company : GC Armor
Stuff handed out during class quests : Artifact Armor
Stuff from Wanderer's Palace : Palace Armor
Stuff from Amdapor Keep : Amdapor Armor
Stuff dropped from Pirmal fights : Primal Weapons
Stuff purchased with Philosophy Tomes : Darklight Armor
Stuff crafted by DoH : Crafted Armor
**Stuff dropped from Crystal Tower : Crystal-Armor
Stuff purchased with Mythology Tomes : Myth Armor
Stuff dropped from Coil of Bahamut : Allagan Armor

I would be more inclined to call the stuff JSE1 and JSE2 (Job Specific Equipment) than artifact, because they can't be equipped by their native class.