Frankly, though, I'd prefer no one ever used the "conventional language of probability", because it leads to precisely these miscommunications.

If the question had been phrased "Out of two-children families that have at least one boy born on Tuesday, what proportion have a girl? [on natural assumptions about lack of biases or correlations concerning the distribution of children's genders and days]", would you agree that the answer was 14/27?

That was the question the author intended to ask. The dispute may simply be as to whether the question which the author did ask is equivalent to the above; if that is indeed our only disagreement, we can still investigate that dispute further, if you like. But let's first see if the dispute is linguistic or mathematical.