B-)That would be a foul ball.

(Would be a foul ball even if the ball was not low and outside!);)

I also have a foul ball in this instance.  The batter did nothing wrong. Therefore, there is nothing to penalize him for.

As long as the batter did nothing intentional, it's simply a foul ball.

Yes, it would be a dead ball.

A foul ball is a dead ball.

When this happens immediately at the plate area, before the batter has time to leave the box, it will be a foul ball even if the ball is in fair territory. (such as right over home plate). This is not spelled out in the rule book, but the way it customarily is ruled.

The rules you quote pertain to a ball that (and bat) that meet  "after immediately".

Then the general rule, assuming all are unintentional,  is:

Fair Territory: Ball hits bat - Play on

Fair Territory: Bat hits ball - OUT

Foul Territory: Ball hits bate - FOUL

Foul Territory: Bat hits Ball - FOUL

This goes back to the thought of ruling with the book and not by the book.  Umpires - especially new ones - have to remember that the Official Rules of Baseball put out by The Sporting News is a book about 4 X 5 inches and about 1/4" thick.  The interpretations of those rules fill a bunch of binders about a foot thick.

I got into a whole discussion earlier about what happens when both the ball & bat are moving, e.g. a rolling bat hits a rolling ball.  Now (for me) it gets interesting.  I'd always used as a rule of thumb - moving ball, stationary bat, play on - stationary ball, moving bat, out - moving ball, moving bat, out.  Turns out I was wrong on the last one, it's not so cut-and-dried.  The latest thought I have on the matter is if the moving ball is 'chasing' the moving bat then play on.  If they're on a 'collision course', which is moving faster?  I'm not 100% convinced but that's my story & I'm sticking to it.