Interesting problem.........my son shoots a kreighoff 32 which did a similar thing, only on the bottom barrel, only with new AA winchesters. It is a tube set and it did it with every gauge.
Our problem was head space. It seems the rim of the shell being too thin combined with the excessive headspace caused light strikes. Took it to Kreighoff and they wanted to reset the barrels and reblue the gun, took it to another Kreighoff trained gunsmith and he said there was no headspace problem, took it to a third Kreighoff gunsmith and he wanted to just do an annual and swore that would fix it, right after Krieghoff had done one......time for a little ******* engineering.
Took some colored fingernail polish and built up the recess where the shell rims rest in the chamber. Hasn't had a misfire or light strike in about 50k shells since. Have to use colored polish so you can tell when you need to recoat. Heck of a lot cheaper than any of the offered professional solutions and almost no down time or exhorbitant charges for shipping a gun to and from a gunsmith.
Your's could be anyone of the problems mentioned above........but if the nail polish works, who cares what anyone else thinks? The idea of the problem only happening on the top barrel to start with made me think the gun was not being opened completely, but that's hard to do with the bottom barrel.
Not to be arguementative, but I have had several 680 series shotguns apart, and none have had half cock notches in the hammers, and yes, I do know what a half cock notch is and how to recognize one. Maybe all of mine have been different from those mentioned above. They do vary from almost new to about 18 years old, I have lots of 680 series shotguns, I would think I would have seen this before.
Seeing how it has been to Cole's, It probably is not a drage issue with a dirty action.
Given the work done on this gun....I'd check the seating depth on the primers that were light struck, and if they are okay, I'd try the nail polish thing. The worst it could do is not work and have to be removed with nail polish remover or scraped out and kill 30 minutes.