Let us assume the information posted was accurate and at the point of takedown it was accurate, but the person posting it has, as you said, the full right for takedown. I've posted it accurately. It remains accurate and I want it down. I simply don't want to share that information anymore. However, someone in the meantime has taken that information and put it somewhere else.
What do you see as my right in a situation like that to remove something that I've put up that is and remains accurate?