All civ introductions close with: "Can you build a civilization that will stand the test of time". I take that as the criteria.
My interpretation:
1) Currently exists, strong and globally dominent: America, Arabia, England, France, Germany, Spain
2) Currently exists, weak but regionally significant: China, India, Japan, Mongolia, Russia, Siam
3) Lost as nation-states but remains alive beneath the surface of dominent cultures: Aztec, Iroquois, Ottoman, Songhai
4) Extinct but left a lasting legacy that still affects our daily lives: Persia, Rome
Ones that did not stand the test of time (correct me if I am wrong):
Babylon - well known for their code of laws and mathematics but other civs developed these concurrently and independently. I don't see anything about them that is unique and still affecting us today. To me, they do not stand the test of time.
Egypt - well known for their religious culture but the nation-state Egypt today speaks Arabic and practices Islam. I don't see traces of ancient Egyptian culture existing in an Egyptian's daily life distinct from the rest of the Middle East. Buildings and artifacts in museums tell us what they once were, but again, did not stand the test of time.
Inca - after the Spanish occupation, the ethnic peoples that were once Incans are in scattered villages and their old culture is lost. Even the Mayans are distinct and still have resistance movements for independence, unaware of anything similar for Incans. I am actually undecided for this one.
With Polynesia, I put it in category 2). Clearly deserving a place: that little triangle diagram missed Taiwan and Okinawa.
Unique Unit: Maori Warrior
Unique units should be judged by how well they did against other civs. I am not aware that the Maori were very successful against the English, even relative to other "indigenous" populations, but I could be wrong.
Also, I think a category 2) Central Asian culture is missing. Not the nomadic ones covered by Mongolia, but around Armenia, Azerbaijan, Georgia. I am not sure what the country or leader will be though.