Chester says Doc's gun is a ball and powder percussion pistol, when Matt takes it from him you can clearly see it is a revolver.
The bullet Matt shows is a copper jacketed hollow point. They didn't have these types of rounds in the old west, and BTW Docs gun is a cap & ball percussion revolver. They were used during the Civil War up until the advent of the Colt Peacemaker. They fired a ball projectile and later a projectile that looked more like a modern day bullet but was called a mini-ball.