It is correct that when the first die landed on 2, there is a 1/6 chance that the second die also landed on 2 (6 possible outcomes, 1 success).
It is correct that when the second die landed on 2, there is a 1/6 chance that the first die also landed on 2 (6 possible outcomes, 1 success).

So, naturally we want to combine these probabilities to look at the problem where at least one of the dice landed on 2 - because we know either the first or the second die landed on 2. If we combine the two conditions, we get 2 successes out of 12 outcomes, or 1/6. However, there is an issue when we do so, in that the two events are not mutually exclusive. The roll 2-2 satisfies both conditions that the "first die is a 2" and the "second die is a 2," so when we combine the outcomes, we have double counted 2-2, and that is not okay. So we remove one of the 2-2s that we double counted, removed 1 success and 1 outcome. Leaving 1 success out of 11 outcomes, or 1/11.

1/6 - We know that a specific die landed on 2, what is the probability that the other die landed on 2?
1/11 - We know that at least one die landed on 2, what is the probability that they both landed on 2?