It's good that he is okay, and I am glad you came here for advice. We tend to go on and on with our advice, because each of us has a little more to add.
I have also cut my birds' toes, right down to the quick, and caused much bleeding. Mine are finches, and like yours these little guys can bleed out in a heartbeat so it is really scary and distressing -- for you and them!
Sounds like you did all the right things and the vet agrees so no harm done.
I have decided the problem is that, along with these big clippers we use, here is what happens. You are holding the bird AND the leg AND the clippers, and trying to extend the toe out from the other toes. You need 4 hands at minimum. So you just palm the bird and hold the leg, and you're trying to clip and THEY MOVE and you've clipped what you shouldn't have. Boom! All in an instant.
So from now on, or from six months ago on, I'm holding the toe with my fingers and clipping a hair short of where I really plan to. After all, we can always clip again if needed.
Hope this helps!