Well there are actually four (4) trigger types.
Single, plain triggers...I have these on two flintlocks rifles, and one caplock rifle, as well as on two trade guns and three muskets...and no worries.
Single set triggers, you can use them as is OR you push them forward from behind to "set" them. Modern version may be used set or unset, and older versions, sometimes need to be set prior to going to full-cock. These are quite expensive these days...$90 +.
Single set double triggers. The rear trigger sets the trigger, and the forward trigger fires the lock. These often need to be set before one can fully cock the piece, and thus the shooter is limited to the light trigger pull. Popular with some target shooters.
Double set double triggers. These like the single set use the rear trigger to set the forward trigger, BUT unlike the previous type, one can fire the piece without setting the trigger. Often the trigger pull is quite heavier, but sometimes it's not bad. This final type of trigger allows, as with the first type, the shooter to either set the trigger for a very light pull ...aka a "target shooting pull"...., but also to fire the rifle with the trigger unset, giving a hunter perhaps the option of using a thick glove over the shooting hand, and thus reducing the chance at a premature firing. These tend to be what one finds in factory muzzle loaders sporting set triggers these days, and they are about 1/3 to 1/2 the cost of the single set single trigger above.
LD