You stated that you think that the idea of increasing or decreasing your confidence that a thing will (or did) happen and the idea of increasing or decreasing the probability that it actually will (or did) happen are “the same thing as long as your confidence is reasonable”. I disagree with the idea that the probability that a thing actually will (or did) happen is the same as your confidence that a thing will (or did) happen, as illustrated by these examples:

John’s wife died under suspicious circumstances. You are a detective investigating the death. You suspect John killed his wife. Clearly, John either did or did not kill his wife, and presumably John knows which of these is the case. However, as a detective, as you uncover each new piece of evidence, you will adjust your confidence that John killed his wife either up or down, depending on whether the evidence supports or refutes the idea that John killed his wife. However, the evidence does not change the fact of what actually happened – it just changes your confidence in your assessment that John killed his wife. This example is like the Newcomb example – Omega either did or did not put $1M in the second box – any evidence that you obtain based on your choice to one box or two box may change your assessment of the likelihood, but it does not affect the reality of the matter.
Suppose I put 900 black marbles and 100 white marbles in an opaque jar and mix them more-less uniformly. I now ask you to estimate the probability that a marble selected blindly from the jar will be white, and then to actually remove a marble, examine it and replace it. This is repeated a number of times, and each time the marble is replaced, the contents of the jar is mixed. Suppose that due to luck, your first four picks yield two white marbles and two black marbles. You will probably assess the likelihood of the next marble being white at or around .5. However, after an increasing number of trials, your estimate will begin to converge on .1. However, the actual probability has been .1 all along – what has changed is your assessment of the probability. This is like the smoking lesion hypothetical where your decision to smoke may increase your assessment of the probability that you will get cancer, but does not affect the actual probability that you will get cancer.

In other words, from my point of view, "the probability a thing will happen" just is your reasonable assessment, not an objective feature of the world.

In both the examples listed above, there is an objective reality (John either did or did not kill his wife, and the probability of selecting a white marble is .1), and there is your confidence that John killed his wife, and your estimation of the probability of selecting a white marble. These things all exist, and they are not the same.

Again, every time I brought up the idea of a perfect correlation, you simply fought the hypothetical instead of addressing it.

You brought up the idea of omniscience when you said:

Note that if your estimates are different, you may be certain you will get the million if you take one box, and certain that you will not, if you take both.

and I addressed it by pointing out that omniscience is not a part of Newcomb. Perfect correlation likewise is not a part of the smoking lesion. Perfect correlation of past trials is an aspect of the Newcomb problem, but perfect correlation of past trials is not really qualitatively different from a merely high correlation, as it does not imply that “you may be certain you will get the million if you take one box, and certain that you will not, if you take both” in the same way that flipping a coin 6 times in a row and getting heads each time does not imply that you will forever more get heads each time you flip that coin. I did consider perfect correlation of past trials in the Newcomb problem, because it is built in to Nozick’s statement of the problem. And, perfect correlation of past trials in the smoking lesion, while not part of the smoking lesion as originally stated, does not change my decision to smoke.

I was not fighting the hypothetical when I stated that omniscience is not part of Newcomb – I merely pointed out that you changed the hypothetical; a Newcomb with an omniscient Omega is a different problem than the one proposed by Nozick. I am sticking with Nozick’s and Egan’s hypotheticals.

It is true that I did not address Yvain’s predestination example. I did not find it to be relevant because Calvinist predestination involves actual predeterminism and omniscience, neither of which is anywhere suggested by Nozick. In short, Yvain has invented a new, different hypothetical; if we can’t agree on Newcomb, I don’t see how adding another hypothetical into the mix helps.

I have stated my position with the most succinct example that I can think of, and you have not addressed that example. The example was:

Suppose that 90% of people with the lesion get cancer, and 99% of the people without the lesion do not get cancer.

Suppose that you have the lesion. In this case the probability that you will get cancer is .9, independent of whether or not you smoke.

Now, suppose that you do not have the lesion. In this case the probability that you will get cancer is .01, independent of whether or not you smoke.

You clearly either have the lesion or do not have the lesion. That was determined long before you made a choice about smoking, and your choice to or not to smoke does not change whether or not you have the lesion.

So, since the probability of a person with the lesion to get cancer is unaffected by his/her choice to smoke (it is .9), and the probability of a person without the lesion to get cancer is likewise unaffected by his/her choice to smoke (it is .01), then if you want to smoke you ought to go ahead and smoke; it isn’t going to affect the likelihood of your getting cancer.

A similar example can be made for two-boxing:

You are either a person whom Omega thinks will two-box or you are not. Based on Omega’s assessment it either will or will not place $1M in box two.
Only after Omega has done this will it make its offer to you.
Your choice to one-box or two-box may change your assessment as to whether Omega has placed $1M in box two, but it does not change whether Omega actually has placed $1M in box two.
If Omega placed $1M in box two, your expected utility (measured in $) is: $1M if you one-box, $1.001M if you two-box
If Omega did not place $1M in box two, your expected utility is: $0 if you one-box $1K if you two-box
Your choice to one-box vs two box does not change whether Omega did or did not put $1M in box two; Omega had already done that before you ever made your choice.
Therefore, since your expected utility is higher when you two-box regardless of what Omega did, you should two-box.

I don’t know that I can explain my position any more clearly than that; I suspect that if we are still in disagreement, we should simply agree to disagree (regardless of what Aumann might say about that :) ). After all, Nozick stated in his paper that it is quite difficult to obtain consensus on this problem:

To almost everyone it is perfectly clear and obvious what should be done. The difficulty is that these people seem to divide almost evenly on the problem, with large numbers thinking that the opposing half is just being silly.

Also, I do agree with your original point – Newcomb and the smoking lesion are equivalent in that similar reasoning that would lead one to one-box would likewise lead one to not smoke, and similar reasoning that would lead one to two-box would lead one to smoke.