Thank you. This is probably the first sensible post from those who are in the 1/11 "camp."
Honestly, I don't know how to phrase the question so that the appropriate answer would be 1/11, but I am sure there is a properly phrased question that would elicit that answer as the correct answer.
I can only say this:
When someone tells me that there are two dice, and at least one die shows a 2, then common sense and my knowledge of what is on a six sided die tells me that the chance of a 2 showing on the "other die" has to be 1/6. To put it another way (which I have said before) the information provided to me only allows me to answer 1/6. The answer is defined by the question.
That doesn't exclude 1/11 as being an answer -- it's just that it can't be the answer given the information that was provided in the stated original question.
Thank you again for asking this.