Outside lane to 1st base - Help

[kycat1]

I don't have my book with me but I am pretty sure I got this correct. If not someone let me know and tell me rule numbers.

ASA slow pitch - Batter hits slow grounder to pitcher. Batter / runner assumes pitcher will make a good throw to 1st and after about 1/2 way to 1st base starts to walk to dug out that is on the 1st base side. Never goes into the dugout but is way out of 1st base line on the foul side. Pitcher flips ball way over first basemen's head and runs after it. Batter / runner sees the bad throw and then runs to first from near dugout (NEVER IN DUGOUT). Batter / runner beats the first basemen to the bag. I call him safe and the defense said he was out of the 2 lines while running to first when there is a play at first and should be called out. I say NO. He only has to run in between the lines if there is a throw from near the foul line or an area where he might interfer with the play. I said he can run outside of the line in any kind of zig zag he wants to as long as he beats the throw and does not interfer with the throw.
I am correct am I not????
Thanks