Lathe stuff:
Swing- the total dia. of work that can spin; ie, center to bed X 2 (you absolutely gotta clear the bed!) You rarely will run stuff that big, probably never in gun work.
Distance between centers- just that; sort of indicates the max length of work, but not much is really done just between centers. With a chuck on it takes up space, though the smaller dia. muzzle can often extend into the spindle bore. With the breech end supported by the steady rest you have to have room to run the carriage and/or the tailstock with tooling mounted, all which takes up space. So, you usually want to get as much barrel into the spindle as possible to maximize the length of bed to work in. It is real awkward to be hanging way off the tail end trying to work something that is just on the edge of too long. With the breech too close to the chuck to use the steady you need to have a spindle 'spider' off the far left end to center and support the muzzle end of the barrel.
BTW, you can make ball bearing 'fingers' for the steady or a 'spider' collar to center around the barrel to run the brass fingers upon. You might as well figure you will mar a barrel running either fingers on it directly.
Most of your work will take place in no more than 4" off the chuck, usually less.