According to Moore, in the first case there's a mistake. In the second case, there is no mistake. — Ciceronianus the White
Well, Moore is dead wrong. He offers an example of one being mistaken and another pointing that out, while it's happening, and then goes on to ask why one cannot say the same things about themselves, which is to ask why someone cannot point out their own mistake while it's happening.
That is exactly what "It's raining, but I do not believe it's raining" is doing(would be doing anyway) when one is pointing out their own mistakes from afar, like another.
Aside from this, I think that you and I are largely on the same page. The statements about the weather and my belief are true(if it's raining but I do not believe it) when spoken by another, but they are self-contradictory whenever they are spoken by me(the mistaken one).
Self contradictory statements are not true, cannot be true. They are meaningful. They must be. Contradictory statements cannot both be believed at the same time. They can be reported upon, shown as such, but they cannot both be believed at the same time. In order to report upon our own lack of true belief about the weather like another does, we must point out the mismatch between what's going on, and the fact that we are oblivious to what's going on; those facts/events/happenings/ongoings/etc.
If we are oblivious, and we certainly are during such times, then we cannot report upon that because there's no difference, in our own minds, between what's going on and what we believe about that. That's why they cannot both be believed.
When viewing a visual recording of ourselves being unexpectedly surprised by rainfall, we'll watch what others did at the time it actually happened. We'll watch ourselves walk right past the coat closet. Yep. We did not believe that it was raining outside. That's where we keep our umbrella.
Hanging on a hook securely attached to the outside of the sidewall. We walked right past. From our vantage point, we can know with utmost certainty that we did not believe it was raining outside.
The difference between the two accounts of our own mistakes, one being reported by another, and the self-reporting, is that another could inform us of the weather outside ensuring that we grabbed our umbrella thereby successfully avoiding unexpected rainfall.
But we cannot convince ourselves to grab the umbrella out of the coat closet on the way out the front door. That is true regarding both timeframes, now and then! It's also true at each and every individual point in time in between...