When folks talk about "barrel runout" and it being an important issue many of them are thinking of some of the old barrels made by Numrich, Douglas and a few others that were very bad.
I'm talking 1/8 inch or more.
There was a post a while back that showed one of these old barrels and try as I might, I can't find it. Anyway, the bore was off center by over 3/16 of an inch as near as I could eyeball the picture.
Now, THAT'S runout!
That's also enough to make it important to install the barrel with the error to the top or the bottom.
That barrel by the way was so far off that I suggested that the it be installed with the thick side at the breech "up" simply because cutting the rear sight dovetail could have seriously weakened the barrel.
Anyway, all of the currently made barrels that I've seen (Green Mountain, Coleran, Rice) are well made and the runout error cannot be seen with the naked eye. Barrels that are that good can be installed which ever way the builder wants to install them without affecting the shooting of the gun.