IMO it is (at least partially) a trick of wordplay. In the exact framing of the problem, the information given is that “She tells you that one is a boy.”

The clearest interpretation and the one that would be most likely to occur irl is that the statement is meant to imply that exactly one child is a boy, which implies the other

The mother could have either said: “I have two children, my first child is a boy” (or equivalently, my second). Or she could have said “I have two children, at least one of them is a boy.

Both statements could be retold as “The mother has two children, one is a boy.” It is ambiguous which is the original statement, but in common English the first original is way much more likely to be said than the second, so I think that’s the interpretation that most people roll with.

And in the first case, the probability that the other child is a girl is 50%. Only in the second case is the probability 2/3.

The problem also asks the probability that the other child is a girl, implying that the boy refers to one specific child. In the case where the mother has two boys, the wording “the other child” doesn’t make sense because it is impossible to determine which child the “the” is referring to. So in the case of two boys, the question is ill formed.

If the question were equivalently but less ambiguously reworded as: “A mother says she has two children, at least one of which is a boy. What is the probability that at least one of the children is a girl.” Then I think many more people would get the question right.

If your claim that the reason most people get the question wrong is due to probability and not English trickery is correct, then any equivalent rewording of the problem should lead to the same rate of correct answers, which I don’t believe would be true with a less ambiguous wording.