Fair ball in front of the plate anc contact with bat on the ground

Question
Situation - the batter swings at the ball which turns out to be slow roller towards first base.  As the batter runs towards first, she lays the bat down in fair territory on the first base side.  The ball rolls into and hits the bat which is in fair territory and the balls stays in fair territory.

Is this interference (batter out) or is the bat then considered part of the playing field?  In this case, the batter beat out the throw and was call safe.

Would the call be different if the bat was in foul territory and causes a ball in foul territory to go fair?

Answer
This is a subjective call.  You have two options and you know it when you see it.  Did the ball hit the bat or did the bat hit the ball?  It sounds like the ball hit the bat in your scenario, this is nothing, play on.  If the bat hits the ball, dead ball batter out for interference, everybody else return to their base.

If all this happened in foul territory, in both scenarios you just have foul ball.  Once the ball hits the bat in foul territory the ball is foul and dead.  You can not have a fair ball in this situation.

Tom