1. The first intuition is that the counterfactual involves changes the physical result of your decision making, not the process of your decision making itself. The second intuition is that the counterfactual involves a replacement of the process of your decision making such that you'd take another action than you would normally do.
2. I imagine it as the following:
   • Physical intervention: I imagine that I'm possessed by a demon that leads me to take the physical actions to choose another option than I would have voluntarily.
   • Logical intervention: I imagine that I was a different person with a different life history, that would have led me to choose a different path than the me in physical reality would choose. This doesn't quite communicate how loopy logical intervention can feel, however: I usually imagine logical alternative futures as ones where you effectively have 2+2=3 or something equally clearly illogical as a part of the bedrock of the universe.
3. I don't think that different problems lead one to develop different intuitions. I think that physical intervention is the more intuitive way people relate to counterfactuals, including for mundane decision theory problems like Newcomb's problem, and that logical intervention is something people need clarifying thought experiments to get used to. I found Counterlogical Mugging (which is counterfactual mugging but involves a statement you have logical uncertainty over) as a very useful intuition pump to start thinking in terms of logical intervention as a counterfactual.

For a more rigorous explanation, here's the relevant section from MacDermott et al., "Characterising Decision Theories with Mechanised Causal Graphs":

But in the Twin Prisoner’s Dilemma, one might interpret the policy node in two different ways, and the interpretation will affect the causal structure. We could interpret intervening on your policy ˜D as changing the physical result of the compilation of your source code, such that an intervention will only affect your decision D, and not that of your twin T . Under this physical notion of causality, we get fig. 3a, where there is a common cause S explaining the correlation between the agent’s policy and its twin’s.

But on the other hand, if we think of intervening on your policy as changing the way your source code compiles in all cases, then intervening on it will affect your opponent’s policy, which is compiled from the same code. In this case, we get the structure shown in fig. 3b, where an intervention on my policy would affect my twin’s policy. We can view this as an intervention on an abstract "logical" variable rather than an ordinary physical variable. We therefore call the resulting model a logical-causal model.

Pearl’s notion of causality is the physical one, but Pearl-style graphs have also been used in the decision theory literature to represent logical causality. One purpose of this paper is to show that mechanism variables are a useful addition to any graphical model being used in decision theory.

There's a range of interpretations for any counterfactual. One must open up the "suppose" and ask, "What am I actually being asked to suppose? How might the counterfactual circumstance have come to be?" We can accordingly do surgery on the causal graph in different places, depending on how far back from the event of interest we intervene.

To make X counterfactually have some value x, we might, in terms of causal graph surgery, consider do(X=x). Or we might intervene on some predecessors of X, and use do(Y=y) and do(Z=z), choosing values which cause X to take on the value x, but which may have additional effects. Or we could intervene further back than that, and create even more side-effects. We might discover that we are considering a counterfactual that makes no sense — for example, phosphorus matches that do not burn, yet human life continues.

In Newcomb's Problem, the two-boxing argument intervenes on both the decision of the person faced with the problem, and Omega's decision to fill the other box or not, as if there were hidden mechanisms that were to pre-empt both decisions in each of the four possible ways they might be made. (This obviously contradicts one of the hypotheses of the problem, which is that Omega is always right.) The one-boxing argument intervenes on the choice of policy that produces the subject's decision, and does not intervene on Omega.

I could call these CDT and FDT respectively, except for the tendency of people to modify their preferred decision theory xDT in response to problems that it gets wrong, and claim to be still using xDT, "properly understood". I just described the one-boxer's argument in causal terms. That does not mean that CDT, "properly understood", is FDT.

ETA: While googling something about counterfactuals, I came across Molinism, according to which God knows all counterfactuals, and in particular knows what the creatures that he created would do of their own free will in any hypothetical situation. Omega is probably an angel sent by God to test people's rationality. (Epistemic status: jeu d'esprit.)

My understanding is that the two contradictory theories are causal decision theory (CDT), which says to choose the action that will cause the best consequences, and evidential decision theory (EDT), which says to choose the action such that the consequences will be best conditional on the action you chose. Newcomb's problem makes causal decision theory look bad but evidential decision theory look good. (CDT two-boxes because your choice seemingly can't cause the prediction, but EDT one-boxes you have more money conditional on one-boxing.) But the smoking lesion problem makes evidential decision theory look bad and causal decision theory look good. (CDT gets to enjoy smoking because it doesn't cause health problems according to the thought-experiment setup, but EDT doesn't because according to the setup, you have health problems conditional on enjoying smoking.)