My point is that there is nothing wrong with Jeff's question, and I believed that you made a distinction with the Monty Hall version. But your second algorithm isn't the same as Jeff's question, my mistake.
It is very clear that my argument specifically adresses the question the way Jeff has formulated it, so you can even forget the part about "your second formulation", what I meant was the question of Jeff and why there is nothing wrong with it.
In statistics, it is commonly accepted that we don't know what we don't state (for an obvious practical purpose). Therefore, when we say : "One of them is a girl", it is implied that we don't know wich one because we didn't specify it.
Therefore there is nothing wrong with Jeff's question (and the probability is 2/3), at least for anyone familiar with basic statistics and probabilities conventions (and I suspect you are), anything else is just quibble. I assumed your argument wasn't about nitpicking this kind of convention, but it turns out I was wrong. I guess nobody wins this one :)