Why? Why do you persist in considering the various faces on both dice? When "at least one die" is identified as showing a 2, that die's other faces cannot be considered.
And by the way, this is the same problem with how the "textbook version" of the question is written. It is also poorly written, and should be changed. But, I understand how the "textbook version" will have the answer of 1/11 because it was written in the context of a probability lesson. Unfortunately, how it was written as part of a probability question is not good enough for a normal conversation. I realize you don't understand what I am saying or what redietz is saying or what regnis is saying about how sentences are written but we deal with "words" and you parrot math lessons. This is why I also wrote on the Wizard's forum that I wouldn't hire anyone for my company who says the answer is 1/11 -- I don't need that kind of overthinking of a basic problem.