This is my understanding:
"Organic" doesn't mean anything in terms of how chickens are kept or where they live. They are fed organic grain. That's it. Not great from a nutrition of the egg standpoint AIUI.
"Cage free" means they are kept inside all the time, just not in tiny individual cages. I think this usually means largish roosts. No outside time, no benefit from a TF food value standpoint. Nicer for the chickens than spending their lives in a box, but it doesn't help the egg.
"Free range" means the chickens can move around freely, but not necessarily outside. It usually means that they are kept in a big barn, with room enough to walk around, and have "access" (tiny doorway) to fresh air (small porch). I've read that farmers often do things when raising the chicks that make them afraid to use the doorway to the outside. So they're still fed 100% grain/soy, they get no sunshine, no grass, and the eggs have none of the TF nutritional benefits.
"Pastured" (not sure if this is an official term) means that chickens spend at least some of the daylight hours outside, hopefully on grass, eating grass and bugs and supplemented with chicken feed (organic or not). That's what makes the yolks yellow, improves the omega 3 content, puts the CLA fatty acids into the yolks, and causes the hard strong shells.
What's worse, non pastured egg farmers often add coloring to the chicken feed to darken the yolks, to make it look like their eggs are pastured.
