The drill command refers to which flank the formation is dressing on. Usually it is "by the right" or "by the centre". You'd typically only dress on the left when left wheeling etc.
The directing flank, when a formation is marching in column of threes, is that flank which would have constituted the front rank when the formation is in line.
Imagine: a formation of men in three ranks standing in line. For purposes of illustration there are 30 men, so there are ten files of three, viz. a frontage of ten men.
The formation is given "Right Turn" - the formation is now in column of threes and what was the front rank of ten men is now on the left vis a vis the direction the men are now facing, and so normally that will be the directing flank, and the order to march will be preceded by, "By the Left." If they had turned left, then the directing flank would be on the right, etc.
If the marching column (now dressing by the left) is given the command to about turn on the march, then the directing flank is now on the right, and immediately after executing the about turn manouevre the command "By the Right" is given.
The command to dress by the centre is an option when a formation advances in extended line, rather than in column.
None of which explains why a formation always steps off with the left foot ! It has to be one or the other - maybe some Roman centurion flipped a denarius, heads for right foot, tails for left . . . . . who knows ?