The craps table example - where one die lands on 2 and the other die is hidden - is not equivalent to the question posed. Once you know for certainty the value of a specific die is 2, then the probability of 2-2 is 1 in 6.

The question posed is "at least one of the dice is a 2." For this question, we do not know if it is the first die that is a 2, or if it is the second die that is a 2 (for how ever you want to define "first die" and "second die"). The person who viewed the dice knows this information, but we do not. This is where the 1 in 11 answer comes from - there are 11 ways to roll at least one 2 (2-1, 2-2, 2-3, 2-4, 2-5, 2-6, 1-2, 3-2, 4-2, 5-2, 6-2), and each of these has the same probability. Of these 11 ways, only 1 way satisfies the condition that "both dice are showing a 2." Hence, 1 in 11.