Will's link has an Asimov quote that supports the "self-image vs right answer" idea, at least for Asimov:

I would, without hesitation, take both boxes . . . I am myself a determinist, but it is perfectly clear to me that any human being worthy of being considered a human being (including most certainly myself) would prefer free will, if such a thing could exist. . . Now, then, suppose you take both boxes and it turns out (as it almost certainly will) that God has foreseen this and placed nothing in the second box. You will then, at least, have expressed your willingness to gamble on his nonomniscience and on your own free will and will have willingly given up a million dollars for the sake of that willingness-itself a snap of the finger in the face of the Almighty and a vote, however futile, for free will. . . And, of course, if God has muffed and left a million dollars in the box, then not only will you have gained that million, but far more imponant you will have demonstrated God's nonomniscience.9

Two boxing is relying on your own power to think and act. One boxing is basically knuckling under to the supposed power of the predictor. I will trust that I am helpless before him. He is an agent and I am his object.

Theists one box and atheists two box. Not surprising.

The twist here at LW is that lots of people seem terribly enamored of these recursive ill posed problems, and think they have solutions to them. That's where the one boxing comes from here, IMO.