I’m going to approach this from a slightly different perspective than Manfred, which may or may not help at updating your intuitions. (It boils down to the same argument, but with less example and more terminology.)
The stopping rule gives you no information above the full data. That is, not “12 heads and 8 tails”, which is just a summary statistic, but a twenty character string of “T”s and “H”s. The stopping rule can give you information about the full data, given the summary statistic. If I know that Statistician 2 got 12 heads and 8 tails, I can rule out some of the sequences of 20 flips that have 12 heads and 8 tails, because I know that statistician 2 never would have gotten them, because they would have stopped midway.
(This is what Manfred illustrated by highlighting that the only possible sequences are THT and TTH, even though HTT has the same summary statistic.)
I’m going to approach this from a slightly different perspective than Manfred, which may or may not help at updating your intuitions. (It boils down to the same argument, but with less example and more terminology.)
The stopping rule gives you no information above the full data. That is, not “12 heads and 8 tails”, which is just a summary statistic, but a twenty character string of “T”s and “H”s. The stopping rule can give you information about the full data, given the summary statistic. If I know that Statistician 2 got 12 heads and 8 tails, I can rule out some of the sequences of 20 flips that have 12 heads and 8 tails, because I know that statistician 2 never would have gotten them, because they would have stopped midway.
(This is what Manfred illustrated by highlighting that the only possible sequences are THT and TTH, even though HTT has the same summary statistic.)