Agapit, I love your posts.
I agree completely that there is a wide range of grip widths that will work, each with its own pros and cons. I favor something in the middle based on the physical dimensions of the vaulter and the position in which the grip will put him or her at the plant. Here is the method I have used for years, and it is very close to the method Agapit suggests to achieve a free takeoff. It has the advantage of being something that an athlete can do at the back of the runway without the need for distracting markers on the pole.
I always found when I used marks that I was constantly worrying about whether my hands were exactly on the lines. I also found it helpful to alter my grip slightly under different conditions within the course of the competition. Exact lines made this more difficult because instead of feeling out the grip I was, once again, worrying about the lines. I hit upon this solution and made it a part of my prejump routine.
This sounds complex and involved when written out, but it is a simple thing to incorporate into a prejump routine with a little practice, and the vaulter never has to worry if their grip is right.
I stood at the back of the runway in a natural posture with my shoulders squared to the box and my feet even. The pole in my top hand rested on my shoulder, and I snugged it up against my chin as far as comfortably possible. This put my top hand just in front of a line drawn from my forehead down. It also naturally placed my elbow to the side. I placed the pole tip on the runway at a point even with a line drawn straight out from the opposite foot. Since I am right handed, this meant that the tip was even with my left foot. I made this position exact by looking at the lines on the side of the runway. If my foot was six inches in from the line, I placed the pole tip the same distance away. That way, even though it might have been sixteen feet down the track, it was still in an exact relationship to my left foot. This posture created a situation where all of the variables were the same. Then I simply reached out with my bottom hand as far as my arm would extend without moving my shoulders. The point where the webbing between my thumb and index finger touched the pole was the optimum grip. I could vary it about two inches in either direction – for varying reasons and with different results, but that is another discussion - without radically changing the timing and basic shape of the vault.