You picked up on a really important issue that is often assumed. Thank you for not letting me just slip it under the radar without question.
To answer your question: we treat the MF and FM differently, just for ease of calculation.
What we're doing is taking a set of possible situations (different 2-child families), each with an attached probability (I'll come back to this), throwing away those that don't match some known information (i.e. those without a boy), taking those that do, grouping them into sets (the set of families with the other being a girl, and the set where the other is a boy), and calculating the probability the actual scenario was in each of those sets.
Now the 'attached probability' probably sounded odd in there, because that's not what we've done, right? We've just been counting situations. True, but that's because we carefully started with situations that are all equally likely. So rather than bugger about with those attached probabilities, we could just use raw counts, knowing that it would work.
So, if we start with four possible sexes of two children: MM, MF, FM, FF, those are all equally likely. I'm using the birth order as a way of breaking the two MF cases apart, so I've got two cases that are equally likely, letting me do the counting trick. I can just count the 2 out of 3 remaining situations that match the information, so 2/3. Easy.
But I don't have to do it that way.
Let's say we follow your intuition and start with three possible sex combinations: MM, MF, and FF. Now, those aren't equally likely, right? There are twice as many families with MF children as with MM children. So I need to use the attached probabilities.
MM (1/4 of 2-child families), MF (1/2), FF (1/4)
We can remove the FF as inconsistent with the information (as before), then the probability the parent has an MF family (i.e. their other child is a girl) is
1/2 over (1/2 + 1/4) = 2/3
as before.
In fact, for any reasonable calculation based on data, we're almost certainly not going to be able to use the counting trick. In the case where I was using actual birth rate statistics for sex (rather than assuming as many girls as boys are born), I'd have to do it this way. The counting trick is really only useful for toy problems and teaching basic probability: you were right to call me out on it.
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The Tuesday's child doesn't drop a case. There are certainly seven cases where, in an MM family, the eldest is a Tue child, and seven where the youngest is, but one of those cases is shared: a case where both are Tue children.
In terms of the intuition you can think of it this way, identifying by birth-day-of-the-week is identifying which child is which, so we're almost at 1/2, but there is one case (both Tue boys) where this information does nothing to distinguish them, so that tiny bit of ambiguity remains, and it can't quite reach the full 1/2, there's still a trace of the 1/3 result that came when the ambiguity was total.