I don’t understand this reasoning. If at least one is a boy, the only configurations I can think of is 1 boy 1 girl or 2 boys. Where does the 1/3 come from?
0 boys - 1/4
1 boy - 1/2
2 boys - 1/4
With the risk of being accused of binarism, there are four distinct possibilities with (close to) equal a priory probability of 25%: older boy/younger boy, older boy/younger girl, older girl/younger boy, and older girl/younger girl.
Discarding the girl/girl case leaves three equally probable cases.
"I have two children, Michael and Alex. Michael is a boy. What's the probability of both being boys?"
If you make a truth table with names as columns, you clearly have only two possibilities for Michael=1.
However if you pick older/younger again you're back to 3 possible states.
I think the answer is still 1/3, but it's a trickier one to reason about immediately.
It seems the question adds information by naming the children, but there's a hidden statement in the form "at least one of them is Michael", which invalidates a truth table with names as columns.
I can only conclude that birth order is an underlying property of the entity. A strict, real differentiator as much as sex is. Names aren't, so names don't add information in this case.
Is there a term for that? Or am I just wrong?
In your variant you need additional assumptions. Will the person always tell you the sex and name of the eldest? Or the names of the boys?
“Michael is a boy” is not really different from “the youngest is a boy”. The probability of both being boys depends on why are you being told that.
[1] Depending in the context the assumption may not be appropriate (a extreme example may be China).
And to count how many ways the universe can give rise to the unordered data sets, the usual technique is to expand the unordered data sets into all the equivalent ordered data sets, and count the latter.