I like the explanation about the coins as well, made it click for my friend when I reworded it that way. "At least one" makes it obvious we don't know which of the coins we are talking about, but it's the exact same thing as saying "One of the children is". You are conveying exactly the same amount of information.
The notion that even if we're not describing the two coins, they are still two separate, unique instances takes a while to get used to, hence the eye-color/age example. In the same way, lets say one of the two coins is my lucky coin and I always flip it first.
If i say at least one of the coins is Heads, I could have gotten lucky coin Heads, and then normal coin Tails. That's HT. Or I could have gotten lucky coin Tails and normal coin Heads. That's TH. Or both Heads, that's HH.
My friend has two normal coins he flips. He says that he also got at least one Heads. Even if he just have two normal coins, they are still two separate coins, in the same way my lucky coin and normal coin is. He still might have gotten HT, TH or HH in the exact same way I did.
Just because one of my coins is my fancy favorite coin doesn't mean the math suddenly changes when comparing probabilities to my friend. We don't know if the first coin landed Heads or the second one did. So it might just as likely be HT or TH.
In both cases we have the same three possibilities (HT, TH and HH), and only one is what we are looking for (HH). 1/3.
For it to be 50% chance, we have to know if it's HT or TH so we can exclude one. With my lucky and normal coin we might say
"Well either the lucky coin is Heads and the other Tails OR the lucky one is Tails and the other is Heads OR they are both Heads, so 1 out of 3".
Comparing it to my friend it might feel weird to say
"Well either on is Heads and the other Tails OR one is Tails and the other is Heads". But it is still correct, my friend is in the exact same situation as I am and we're both asked the same question.
Changing the question to be about two children instead of two coins doesn't change anything. If we say 50% we HAVE to know which child we already know the outcome of.
It still reads "Well either one is a girl and the other is a boy OR one is a boy and the other is a girl OR they are both girls, so 1 out of 3".
Both "Well either one is a girl and the other is a boy OR one is a boy and the other is a girl" and "Well either one is Heads and the other Tails OR one is Tails and the other is Heads" feels weird, but they are correct in both situations.