In both of your koans, the total number of white liberties is one more than black’s. Perhaps there’s something to that.
It holds true with these as well:
However, here it’s reversed:
Also doesn’t explain boards with only one color, so I’m probably grasping at straws.
Conjecture: Rule is color inversion invariant.
True.
Well, I think I’m back to square one. 
I had a direction I was working towards, but then I decided to just throw down random stones and see if I could find any random green koans that broke my conjecture when this happened:
Here’s a small hint:
Here’s a big hint:
Rule guess:
Each chain of stones must have a different number of liberties.
Close! All green boards do satisfy this, but there are also red boards with this property:
I’m having a hard time putting it in words, but is it something like: each additional chain of stones added to the board must have exactly one more liberty than the chain with the current highest liberty count?
I think you pretty much got it! The liberty counts have to form a range of consecutive integers, so for instance
0, 1, 2, 3, 4
or
2, 3, 4, 5, 6
or
2, 3, 4, 5, 6, 7
(it’s quite funny that you found that board “by accident”!)
Should we start the next game?
Are we gonna start a new game?
Anybody is free to start a new game if they want to! I’m a bit busy with other things at the moment so probably won’t be able to commit much time to it right now unfortunately.
Perhaps a game with strings of emoji as Koans instead of boards would be more convenient to set up and play?
For instance, valid koans could be any sequence of these four emoji:
Then it’s more “Zendo” than “Go Zendo”, but after many games with go boards I think some variety would be good anyways 
Yes, let’s play Emoji Zendo. I’m thinking of a rule that specifies a subset of short sequences of emoji (three or less of the emojis built into Discourse). The above is a valid (“green”) example. These are also green:
These examples are red:
Just to clarify, are these sequences or sets of emoji? (i. e. does order matter and are repeats allowed?)
Also, is the name of each emoji significant in some way, or only the image?
I think allowing all possible emoji makes it difficult to make a completely objective rule (of course the rule doesn’t have to be 100% well-defined, I just prefer it that way personally). That said, here’s my best rule guess based on the examples above:
All of the emoji in the sequence have to contain a circular/spherical shape. (the tangerine is not perfectly spherical, but still much more so than for instance )
Maybe the rule refers to the categories of the emojis?
Yep, you got it!
Ok, even though I said I can’t commit much time right now, I guess I can make one quick rule 
Valid koans are sequences (of any length) of red, blue, yellow and black.
These follow my rule:
These do not follow my rule:
Sorry to interrupt…
I feel the tool may be useful in general for the forum, if it could be modified to allow varying board sizes.