I asked about these differences in my second post in this post tree, where I explained how I understood these counterfactuals to work. I explained as clearly as I could that, for example, calculators should work as they do in real world. I did this explaining in hopes of someone voicing disagreement if I had misunderstood how these logical counterfactuals work.
However, modifying any calculator would mean that there can not be, in principle, any “smart” enough ai or agent that could detect it was in counterfactual. Our mental hardware that checks if logical coin should’ve been heads or tails is a calculator the same as any computer, and again, there does not seem to be any reason to assume Omega leaves some calculators unchanged while changes results of others.
Unless, this thing is just assumed to happen, with some silently assumed cutaway point where calculators become so internal they are left unmodified.
Try as I might, I cannot find any reference to what’s canonical way of building such counterfactual scenarios. Closest I could get was in http://lesswrong.com/lw/179/counterfactual_mugging_and_logical_uncertainty/ , where Vladimir Nesov seems to simply reduce logical uncertainty to ordinary uncertainty, but this does not seem to have anything to do with building formal theories and proving actions or any such thing.
To me, it seems largely arbitrary how agent should do when faced with such a dilemma, all dependent on actually specifying what it means to test a logical counterfactual. If you don’t specify what it means, whatever could happen as a result.