For changing subjects, or if your subject leaves the focus area completely, you should release the AF-On button and focus again. But it your subject is moving, simply holding down the AF-On button will do a good job of adjusting focus as the subject moves.
Tracking is slightly different in that tracking relates to objects obstructing focus. If another player quickly passes between you and your subject, AF Tracking can maintain focus on your subject rather than shifting to the passing player. With Focus Tracking Off, AF would leave the original player and try to switch to the new player. Your description of how to use A3 is correct about the lag.
What you don't want to do is continually press and release the AF button every time the subject moves as you will continually be focused on where the subject was rather than where it is at shutter release.