With logic problems the assumption is that logical information is always relevant to the solution, no matter what the source of the information is. It's interesting to have that assumption challenged though.
https://www.youtube.com/watch?v=emiMj8cCL5E and the extra footage is another demonstration on how the solution can change if you don't start with the normal "logic puzzle" assumptions.
the difference here is that the label on the green box is explicitly "challenged" by the label of the red box. the step is to realize that both labels should be challenged (i.e if the label on the green box doesn't have to be true, the label on the red box doesn't have to be true)
I'm not convinced the fact one label "challenges" the other should change how you attack the problem. I think you have to be able to apply the same reasoning to the following problem:
James says he is shorter than Alex.
Jane says she is shorter than James.
Alex says he is taller than James.
Can you figure out who is the shortest?
Going by the authors premise that only things in the puzzle statement can be taken to be truthful, you can't figure out who is the shortest because they could all be lying. It's formulated exactly the same as the problem in the post, it just doesn't have a potential contradiction to lead you down the path to realising you're not meant to be assuming the logical statements are involved in the solution.
If one of the possible answers is "There's not enough information to decide", that's a good hint that something weird may be going on and you should double-check your assumption of "everybody is telling the truth".
https://www.youtube.com/watch?v=emiMj8cCL5E and the extra footage is another demonstration on how the solution can change if you don't start with the normal "logic puzzle" assumptions.