The difference is that I use a 6 o'clock hold with fixed-sight handguns, while my brother uses a bullseye hold.
Not that it matters, but I've always used the terms the opposite way. The 6 O'clock hold is the one bullseye shooters use. Putting the bullseye on top of the front sight makes the position easier to see then trying to put the black front sight right in the center of the black bullseye on the target, plus the black bullseye is a known size at a known distance, so its easy to regulate the sights to hit the center of the target with the top of the front sight on the bottom edge of the bullseye.
Generally speaking, I shoot from a rested position when I want to know what my gun and ammo can do, and an unrested position when I want to see what I can do with my gun and ammo.
I do shoot long barrel Contenders with the gun rested, which is the same as I would do in the field, if possible.
Others I shoot from a rest, with my forearms rested, and the gun held free in my hands. (DO allow enough room for recoil in larger calibers. Trust me, you DON'T want a hard recoiling .44 pinching your hands between it and the rest!
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The "classic" standard of handgun accuracy (for semis & revolvers) is 2-2.5" groups at 25yds. (and just FYI, shooting a .44 or .45, that can be one ragged hole.
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