From the paper:

At START, he is given the option to be woken either at both intersections, or only at X. In the first option he is absent-minded: when waking up, he does not know at which intersection he is. We call the second option ‘‘clear-headedness.’’ As in the previous discussion, the question at X is not operative (what to do) but only whether it makes sense to be ‘‘sorry.’’ If he chose clear-headedness, his expectation upon reaching X is 1/4. If he had chosen absent-mindedness, then when reaching X he would have attributed probability 2/3 to being at X and 1/3 to being at Y.

I disagree. If I opt to be woken at X, then when I'm woken at intersection X, I know I'm at X. Even if the idea is that I'll immediately go back to sleep, forget everything, and wake again at my payoff/destination, I still knew when I was woken at an intersection that it was X.

Unless I chose to be a) woken at X and b) have no memory of having chosen that, in which case it's fair to say that "I'm at some intersection" (but they suppose I remember the road map I was given indicating intersectons/payoffs X,Y). I guess that's what must have been meant.