Dead Ball?

[Rufus]

Situation: 1st base open, 2 strikes on the batter, no outs. Batter swings at pitch and misses for strike three, but the ball strikes him in the arm and deflects away from catcher. BR advances to 1st on the dropped third.

Reading Fed 5-1-1 it says:

Ball becomes dead immediately when: a pitch touches a batter or his clothing; 1. The ball becomes dead even though the batter strikes at it

Fed 7-2-1(b) says:

A strike is charged to the batter when: a pitch is struck at and missed (even if the pitch touches the batter)

Based on the above. Should this have been a dead ball, batter's out on swing/miss on 3rd strike? OBR 6.05(f) seems to address this as well.