@mikeazo, I'm playing with definitions here so you could imagine various versions, but as I was thinking of it I'm allowing Alice to know whatever, including who ends up receiving the bit, so long as the transfer remains "party independent".

@Joseph Sible, yes, in the context of a classical bit commitment scheme that seems right to me, which is an interesting point. That said, I'm working in a context where security should be based on physical restrictions only, ie those coming from quantum mechanics and special relativity. In that context we don't have bit commitment in the hand-commitment then hand reveal sort of form.