the reason your player stops could be because once the catcher throws, they may be throwing to you vs "leading" you to the bag. so you wouldn't want to run past it?
I usually have the mind set that if their is a runner on first and the situation calls that i will be covering the bag, i just assume it's a steal and start towards second the minute i the pitcher move. (playing as SS) I usually get to the base on time, but i always get drafted to a team with a catcher that has a noodle arm, so they never throw anyone out anyways.