Ball hits batter

[Blueplate]

8.1.2.b
Penalty 2
If the batter is hit anywhere on the body,including the hands,while swinging at a pitch,and hits the ball fair or foul,the ball is dead and a strike called. If it is strike 3,the batter is out.

...and hits the ball fair or foul...
It is my understanding that a ball cannot be hit fair or foul unless it has been struck by the bat..and if so then this rule is saying that the bat struck the ball at sometime.

So on 3rd strike, if a ball hits batter then bat, DB,strike,batter out.

But,if a ball hits the bat then batter on 3rd strike,DB,foul ball???
Does that not contradict the rule...ball hit foul or fair...3rd strike batter out???