Runner Abandons Dropped Third Strike

 
The lead-off batter of the inning (B1), swings and misses at an 0-2 pitch in the dirt. After the pitch has hit the ground, the catcher cleanly secures it and starts the traditional "throwing around the horn" as B1 dejectedly walks a long distance to the opening of his first-base dugout. Just prior to entering the dugout, he is told that his time at-bat ended as a dropped third strike, so he runs the short distance to first base and stands on the base.
 
What are the umpire's actions now? Is this a dropped third strike? At what point does a batter abandon his attempt to run the bases?
 
Call
In the situation below, the pitch hit the ground before it was secured by the catcher -- so with the bases empty, the batter is not out until he is put out or abandons his effort to run the bases, In NF, on a dropped third strike, the batter-runner does not abandon his ability to run the bases until he enters the dugout (NF 8-4-1i). So in the situation below, B1 is safe at first base.