I don't think the schism is because of fallible vs infallible so much as people being stuck in naive decision theory (just take both boxes; it's already decided beforehand!) or not (hey, the kinds of decisions I'm willing to make might have changed what Omega decided in the past!).
After all, Omega need not be infallible. So long as he predicts your decision with an accuracy of 50.05% (slightly better than a coin toss), you profit:
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Let p be the probability that Omega predicts your decision correctly.
E(one-boxing) = p⋅$1mil + (1-p)⋅0
E(two-boxing) = (1-p)⋅$1.01mil + p⋅$1k
Solving for E(one-boxing) > E(two-boxing)
p⋅$1mil > (1-p)⋅$1.01mil + p⋅$1k
p($1mil + $1.01mil - $1k) > $1.01mil
p⋅2mil > $1.01mil
p > 50.05%
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And if he doesn't predict you slightly better than a coin toss, why is he called the Predictor?
In that scenario, it wouldn't make a difference what you choose.
If you were so inclined to take Box B's million, Omega would never have put that million in the first place.
The only way for Omega to put that million there is if you weren't inclined to take Box B - despite the million being completely visible to you - in which case you lose anyway.
The situation where Omega gives you the million, and you take it, just never comes up. Can't fool it.
After all, Omega need not be infallible. So long as he predicts your decision with an accuracy of 50.05% (slightly better than a coin toss), you profit:
---
Let p be the probability that Omega predicts your decision correctly.
E(one-boxing) = p⋅$1mil + (1-p)⋅0
E(two-boxing) = (1-p)⋅$1.01mil + p⋅$1k
Solving for E(one-boxing) > E(two-boxing)
p⋅$1mil > (1-p)⋅$1.01mil + p⋅$1k
p($1mil + $1.01mil - $1k) > $1.01mil
p⋅2mil > $1.01mil
p > 50.05%
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And if he doesn't predict you slightly better than a coin toss, why is he called the Predictor?