WOW. How did you, writing geniuses, come to this conclusion? So, to summarize the question as asked, not going to copy for an N-th time, can be simplified to "set a side one die as two and roll the other die"? How exactly was your thought process working in order to convert the OQ into this? Extremely curios ...
Okay, going to copy it none the less. So this:
"You have two 6-sided dice in a cup. You shake the dice, and slam the cup down onto the table, hiding the result. Your partner peeks under the cup, and tells you, truthfully, "At least one of the dice is a 2."
What is the probability that both dice are showing a 2?"
somehow converts to this:
"Set one die on 2, and just roll the second one and make your bets"
What is the relation between those two settings? Anyone? Please help me.