Runners hit by batted balls

[Tabe]

As a bit of an intro:

This is my first post here. I am not an official, nor do I coach anymore. I've come here looking for explanations of rules and things that I can't seem to find anywhere else

That said, the following came up in my nephew's Little League game last night:

Runner on 3rd, two outs. Batter hit a line drive that struck the runner. Runner was either standing on, or jumping into the air above, 3rd base when struck.

Who, if anyone, should be out in this situation?

As it played out, the umpire (a high school kid who was in a little over his head) made no verbal announcement of a call. My nephew's coach asked for the runner to be called out, the opposing team argued the runner was safe because he was physically on 3rd when struck. I then said if the runner was on 3rd that the ball was still live, so the catcher threw to 1st to force the batter. After some discussion, the umpire finally ruled - incorrectly - that the ball was a foul ball and play continued.

I have looked in the official rules on the MLB.com site and I can't find anything that specifically states a batter is out when struck by a batted ball except in cases of interference - which this wasn't - or on an Infield Fly, which also doesn't apply here.

Any assistance, especially one that points to an official rule, would be most appreciated.

Thanks,
Tabe