If we get really stuck and frustrated, I can give some increasingly obvious hints. 
I would like to know about this one please:
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I think the first two we were given are among the most interesting so far:
So let me check if this makes any difference:
Edit: Interesting, both red. So for the first one changing a black chain to white made the board change from green to red.
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What are people’s theories for what might be the important factors? I think we can make more progress if we openly discuss our thoughts and try to explain what hypotheses we are trying to test with various boards.
It seems that liberties, group size, and stone count are not the direct factors, and if it’s a simple rule (as implied) then maybe the rule does not directly use those, even if they may indirectly play a role via some other factor.
I think @le_4TC is on to something with coming back to the first two examples, where it seems that the relative placement of the same shapes is playing a role.
I wonder if the rule has something to do with counting stones along rows, columns, and/or diagonals.
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So far it looks like White and Black colors can be swapped without changing the color of the Koan, but the colors of stones still play a role as le_4TC’s example Koans demonstrated.
It appears as though there are slightly less red Koans than green Koans (with very low certainty, though).
Can anyone find a rule that is consistent with all of the boards so far?
I’ve been trying to think of something simple that works, but keep on getting stuck by finding that a counterexample already exists.
I like the way we’re thinking now, definitely on the right track.
Here’s a bit of a hint:
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May I ask about this one?
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Red 
My thoughts are: RubyMineshaft’s hint implies (possibly among other things), that the position of stones relative to the board is important. I wonder if all Koans change color when shifting the whole Koan one space.
Red indeed.
The one in question is a rotation of an already revealed red Koan.
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I’ve come to the conclusion that not both of the following statements can be true (in other words, one must be wrong)
1.) Swapping colors White <-> Black never changes the color of a Koan.
2.) Shifting the whole Koan one space always changes the color of a Koan.
Otherwise (assuming both are true) one can construct such a counterexample (the following Koans are not submissions, only to demonstrate the argument):
-> Shift -> 
-> Swapping Colors -> 
-> Rotation ->
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I guess I would like to submit this one afterall:
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