The BATTER’S BOX is the area within which the batter shall stand during his time at bat.
Rule 6.03 mandates that the batter's legal position is with both feet within the batter's box. The lines are within the batter's box.
Typically the umpire does not direct the batter to place his foot fully within the batter's box unless it is brought to his attention.
Rule 8.05(e) CMT