ruling on overthrown ball, after first intial play on the batter-runner.

There is no such rule about " abandoning the basepath"!!! The interpretation of "out of base path" 7.08(a) only applys as a tag is being made.

And even though the rule states "direct line between bases" the NAPBL case book instructs umpires to interpret this to mean the direct line between the base runner and the base he's heading towards ... but only as he's about to be tagged...

This can be further illustrated looking at the dropped third strike rule...6.09(b) for reference...

6.09 The batter becomes a runner when_
(a) He hits a fair ball;
(b) The third strike called by the umpire is not caught, providing (1) first
base is unoccupied, or (2) first base is occupied with two out; When a batter
becomes a base runner on a third strike not caught by the catcher and starts
for the dugout, or his position, and then realizes his situation and attempts then
to reach first base, he is not out unless he or first base is tagged before he
reaches first base. If, however, he actually reaches the dugout or dugout
steps, he may not then attempt to go to first base and shall be out.