No, that's just an except from the article about this question:
https://en.wikipedia.org/wiki/Boy_or_Girl_paradox
Your reasoning isn't wrong, you start from faulty assumptions. OP says that "They have two kids, one of them is a girl.". There are three ways to interpret this statement.
First interpretation: 'They have two kids, exactly ONE of them is a girl.' - As you mention, we've been discarding this possibility. (On further review, I think this is the most correct answer based on the exact wording he gave)
Second interpretation: 'They have two kids, one of them IS a girl' - This is how I originally interpreted the question. This is implying that we know that either the older child or the younger child is a girl. The probability that the younger child is a girl given that the older child is a girl is 1/2 (they are independent). The probability that the older child is a girl given the younger child is a girl. The union of these statements is 1/2, that is:
P(Older|Younger) U P(Younger|Older) = 1/2.
Of course this wouldn't depend on age, just that there was some 'measurement' to determine that one of the children was a girl. Here is a way that the OP could have unambiguously phrased his post to make it this option:
"Mr. & Mrs. Teanchi introduce you to their daughter, Susan. They tell you that Suzan has a sibling. What is the probability this sibling is a girl?"
This is clearly 1/2, and if you try to argue otherwise there may be no help for you.
Third interpretation: "They have two kids, at least one of which is a girl" - This is 1/3, by the reasoning you and others gave.
I ranked them in order of my opinion of their validity given the exact phasing of the original question. I'll give the arguments against each of them in reverse order.
3: (1/3 probability) this relies on the question being logically equivalent to "They have two kids, at least one of them is a girl.", but this goes contrary to the stated text of the question "one of them is a girl" and since this question relies on precise language as a trick, I don't think it's fair to assume that means 'at least'
2: (1/2) Probability. This is consistant with the text of the problem as given. Assumes that the phase "one of them is a girl" is describing a distinct child. It does, however, contradict the hint. IMO contradicting the hint is weaker than contradicting the text, as the hint is often given not by the original author, but by the latest teller of the question who may just be wrong.
3: 0. Consistent with both the text of the question and the hint. Kind of a lame answer, but strongest interpretation.