Round two you have the option of passing or swapping. A swap breaks your stun, loses your turn and you bring in a backline pet. Passing loses your turn and you keep your current pet.
Swapping doesn't "break your stun", your pet is still stunned for the round. You just swapped to a pet that isn't stunned.
Passing does lose your turn, but it wouldn't have mattered because you were stunned for that round anyway.
I think what you're seeing has to do more w/ pet speed than the stun itself.
Take your example about the Darkmoon Monkey. If the Monkey was slower than your pet, then you would have already gotten to go in the round that you were stunned.
If your pet was slower than the monkey, you would have been stunned for that round (unable to act) and then also for the following round.
In your example about Magma Trap / Snap Trap, I'm assuming that the pet you were using was faster than the trap using pets (very likely in the case of Magma Trap), meaning that your pet was stunned when you tried to attack and then ate an attack.
On the following round, you continued to be stunned as normal, eating another attack.
Again, passing has absolutely nothing to do w/ "breaking stuns" and never has. You can argue that fact all you like, but that is nothing more than a bit of confusion over the stun mechanics on your part.