The number of "barrels" refers to how many Power Accumulators you are employing in your stroke. Single barrel is one, double is two, etc.
Hitters usually employ a triple barrel stroke using accumulators 1, 2, and 3. A Swinger's triple barrel is usually 2, 3, and 4. There can be four barrel Hitting but not four barrel Swinging.
Do you have the book? 6-B is the key to understanding all of this.