I think I know the answer to your question.
Spoilers below
Only Jack and the blue haired girl can keep their memories after each loop and Jack and the girl goes through the loop in different orders so that's Jack's first loop wasn't a loop the blue haired woman ever experienced, and it was the only loop where someone got voted out during the first vote.
Ohh! That's a good point about the memories. The other part kinda falls apart though
spoiler below...
They go through in different orders, but the loop 0 would be the first instance of the event, so a loop wouldn't have happened yet. I think it would have made more sense if the round 1 elim had have happened a loop or two in.
Nah just hear me out for a second
Spoilers below
I don't think there is a Loop 0. When Jack opens his eyes for the first time the blue haired woman has already been through dozens of loops before him. So even though that's loop 0 for Jack, for the blue haired woman she already been through many loops before Jack and none in those loops did someone got eliminated in the first night which explains why she told Jack that no one gets voted out during the first night.
Basically the whole game isn't really a time loop, it's a spiral.
This should explain what I mean
Also I think the robot girl is the reason Jack got voted out in the first night , she must has rigged the voting just like the blue haired woman claimed.
