Bayesian: And it doesn't bother you that the meaning of your experiment depends on your private state of mind?
Not quite. The private state of mind of the researcher changes nothing. It's only the issue of which question is asked.
In this case, the two questions are (a) what is the probability of such an event occurring after tossing a fair coin 6 times and (b) what is the probability of such an event occurring if a fair coin is tossed until it lands tails.
The meaning of the questions does not change, nor do the answers to them. It is only a matter of what question is being asked -- which is obviously important when conducting a study, but is not so counter-intuitive (and much less confusing) when presented in such way (IMO).