The back action refers to the design of the lock - the metal component you see with the hammer sticking out of it at the front. All the working parts are behind the hammer - hence, the descriptive term 'back action'. It is a percussion lock, BTW, and uses a percussion cap on the nipple that is detonated by the hammer striking it, thus setting off the load of powder in the breech.
The proof marks are on the barrel - you posted the pics yourself.
However - FIRST THING YOU MUST DO is the make sure that the gun is NOT loaded. Get the ram rod and put it down the barrel, and hold it where it stops. Remove it and place it beside the barrel. If it is within half an inch or so of the end of the barrel, you are good to go. If it is a couple of inches or even more short, the you MIGHT just have a load in there - NOT uncommon.
Report back to us here, please, before proceeding.
BTW, 'Old Bermingham' is not a place. The poster was referring to the older style of stamps used by the Birmingham Proof House, replaced a number of times since those shown on your barrels by similar but not identical designs.