When a batter hits a fair ball and their bat strikes the ball in fair territory is the batter is out?

If the batter is still in the batters box, it is a foul ball, otherwise, the batter will be ruled out, and it is a dead ball with runners returning to their bases

This is wrong, if the ball hits the bat a second time in fair territory the batter is out, standing in the batters box means nothing in this rule. see MLB rule 6.05 h