Did AlphaGo and AlphaGo [Zero] play 100 repetitions of the same sequence of boards, or were there 100 different games?
There were 100 different games. You can view some example games between AlphaGo [Lee] and AlphaGo Zero here. They are clearly all different.
This statement in the question shows a misunderstanding:
My understanding of AlphaGo and AlphaGo [...