Bear in mind that the Battlemaster damage maths from the first page is rather bad and somewhat misleading.
It ignores the riders on the maneuvers, to just look at pure damage, but ignores the maneuvers that just give damage: Riposte and precision attack.
A better model would be to assume that each Superiority dice grants an additional attack, rather than just being a rider on an existing attack. This represents use of Riposte to get an additional attack, and Precision to turn a miss into another chance to hit.
This also shows the Battlemaster scaling better since attacks will increase in damage at higher levels.