The simplest case is when a fact that is being considered counterfactually is received from a given observation, so that you can explicitly say where the parameter is in the system, and use the dynamic specification of the system to see what happens to it depending on the parameter. That's the case with the coin and random digit index.

10000th digit of pi is one step more complicated, but it's still independent on most of your knowledge, so it's conceptually easier to localize knowledge about it in your mind. Once you start considering the question, knowledge about its answer starts affecting your dynamic, and this influence can likewise be tracked to the source. That's why I introduced Pi(n) as a local expression: all the knowledge in the algorithm about the answer to this question comes from this single procedure, so by varying its contents you can examine the impact of its different values of the future behavior.

Whether or not 1 is even is much more pervasive, so the surgery that changes it will be hard and not at all intuitively obvious. So, the disagreement seems to be that you trust your intuition about whether it's possible to make 1 an even number in your mind, while I trust the generalization of idea that you can change whether the coin lands on one side or another, whether Pi(10000) is even or odd, and arbitrarily more pervasive questions as well.

This does depend a lot on what Omega understands by the question (how Omega's algorithm logically depends on the question, and on your algorithm), which is related by my unwillingness to conclude that mutual cooperation is the clear-cut outcome of PD. In this thought experiment, this understanding is mostly specified, in other cases intuitive grasp of the problem won't be enough.