I think most recent technique books, including ISTD, IDTA and UKA, as well as Alex Moore 10th Ed, offer all these options. Most describe the Natural Spin Turn entry as starting with step 2 of the V-six, which is why I added the comment that step 4 (the step under discussion) was the one after the LF crosses.
Interestingly, Carl Bryant, in his 1953 book Advanced Ballroom Dancing, describes the V-six starting with step 1 being the man's RF back, which we now regard as step 2. He suggests dancing the figure after step 6 of the Natural Spin Turn or step 4 of the Quarter Turns.