Two children paradox: where is my reasoning wrong? I hope here is the good place to be asking this. Apologies otherwise.

The statement is as follows:

"Ms Michu has two children. We know one of the two is a girl, we call that girl Ludivine. What is the probability that Ludivine has a brother, rather than a sister?"

This is with the usual assumptions that there is an equal chance for one person to be a boy or a girl, no cis or gemels, etc.

I read that the answer should be 2/3. Demonstration is :
For any given set of 2 children, there are 4 different equiprobable combinations:
GG BB BG GB
We get rid of the bb combination because there is no girl. That leaves us with 3 possibilities, of which 2 match the criteria of the statement, hence the probability of Ludivine having a brother is of 2/3.

But, if I follow the following reasoning, I obtain a probability of 1/2:
Ludivine(L) is either the elder(e) or the youngest(y), with an equal probability of 1/2.
If she is the elder(e), it gives 2 combinations, with again a probability of 1/2 for each:
B(y) G(L)(e)
G(y) G(L)(e)
If she is the youngest we again have 2 equiprobable combination :
G(L)(y) B(e)
G(L)(y) G(e)
By combining those probabilities, we have 2 cases out of 4 where she has brother. Therefore she has a probability of 2/4 or 1/2 to have a brother.

Now I feel like my reasoning is wrong somewhere, but I can't see where or why, even though i went through quite a few topics on the subject; hence why I'm re-discussing that old topic...
Thanks in advance!
The problem is not really with your reasoning as much as with your interpretation of the question.

In the first interpretation, we know that there is a girl, and then we pick one of them, and then ask what is the probability that the other child is a boy. In this case, since we know that there is a girl, there are as you say exactly 3 equally probable possibilities, GG,GB,BG. And hence we can check that the answer is $\frac{2}{3}$.

In the second interpretation, we pick a child at random, and then it turns out to be a girl, and then we ask what is the probability that the other child is a boy. In this case, there are exactly 2 equally probable possibilities, GG and GB, where the first letter denotes the gender of the picked child, and the second letter denotes the gender of the other child. Clearly the answer is $\frac{1}{2}$. (Your added "reasoning" about the picked child being elder or younger isn't important.)

So the underlying issue is that when specifying a probability question, it is absolutely crucial to clearly state the order in which actions are performed.
How exactly do we know one of the two is a girl? We don't just abstractly know it. There is some story how we received the information. If we meet a person in hospital visiting a sick child, and that is how we gained the information, there is an equal chance that we would have received the information "Ms Michu has at least one boy" or "Ms Michu has at least one girl". If we meet Ms Michu picking up her child at a girls' school, we couldn't have received the information "Ms Michu has at least one boy" because it's a girls' school and she had to pick up a girl. 

People with two boys turn up at hospitals visiting a sick child. People with two boys don't turn up at girls' schools to pick up their child. So how we got the information influences the result.