Interference
 

Runner stealing 3rd base. Hot ground ball hit up the 3rd base line, runner slides, F6 playing behind the bag, ball hits runner:
#1 before contact with the base while sliding.................Interference!
#2 after contact with the bag, while in contact with the bag, runner still lying on the ground...................Interference?

#1 is actually what happened. I called the runner out....Interference........... I was PU, saw him stealing out of the corner of my eye, so when ball hit, I was looking right down the 3rd base line.........

The base is not a safe haven. You'd have an out in either case.

Under OBR, you get two outs only if the interference was willfull and deliberate with the intent to break up a DP. (oe some wording like that)

What if the runner would have been standing on 3rd base, NOT stealing, just standing there.
Would this be interference or a deadball and BR get 1st?

yes.

Assuming the third baseman is playing behind the base and the runner was hit in fair territory, then you have interference on runner (out), and batter awarded first base.

That's why good coaches teach their runners to stay in foul territory when on 3rd.

Thanks
David

except after the catcher catches the ball...

Yep. Get back in the baseline and don't turn your head away from the catcher.

INT if the batted ball was fair. If R3 is standing on the base and the ball hits him over while it is over foul territory, then it's just a foul ball.