The answer is the same in your coin version. There are four possible outcomes: (Q, D) = HH, HT, TH, TT. Given that one turned up heads that eliminates TT so we see that HH has 1/3 probability.

As you can see there aren’t two HH states just as there aren’t two GG states in the original question.

Well, sorta? You were either in the world where dime was heads, in which case the space has TH, HH, or the Quarter was Heads, in which case you have HT, or again HH. Both with 50% probability, no? You were never in a scenario where HT and TH have the same probability as each other. (Well, again, sorta. Point being that you know one of them is possible.)

Edit: I want to be clear that my initial thinking was exactly what you said. I was trying to "steelman" the bad intuition and I think I've trapped myself. :D