"You can't argue that, because my left foot is 'across' my original direction of travel when the right closes to it, that the original step 2 was across the line of travel."
It's not. However, if I recall correctly the reference was to crossing the LOD, not crossing the line of travel. Since the line of travel is DW, both the first and the second step will quite obviously cross the line of dance.
"So at no time during step 2 is my left foot across my original direction of travel."
Perhaps, perhaps not. One thing to consider is that if there is going to be a turn in the direction of progression during the first half of a natural, to change from the original from DW so that step four can move down LOD, then some force directed towards the center of the room will need to be found. That force would probably come from placing the second step slighly outside the actual path of the body during the second and third steps, so that the body weight can arrive just a cm or two short of that foot and permit a push in this new direction. Fail to do that, and step four would move to DW (as it does on many).
But what is harder to predict is if step 2 will be placed across the original line of travel, or if the line of travel will be modified during step two's rise so that it will pass just short of directly over step two, permitting us to arrest the component of movement towards the wall and send step 4 directly down LOD.