Of course it is a base running mistake by the runner and not the third base coach. The runner needs to know the situation and that his run is meaningless without the runner on second scoring. Anything hit to the left side and he must stay glued to third base. It was also a quickly developing play where the ball was sharply hit to the third baseman. There is not time to obtain coaching input w/respect to whether the runner should break for home or not. This is a decision that must be made by the runner in real-time (and rehearsed in his mind before the play).
Brian