After reading the post by Randy Hees, I believe that he has clearly identified the purpose of the extra head of the Greer spike. It is a necessity that flows from the unique manufacturing process of the Greer spike.
In studying Greer’s patent posted by Sandiapaul above, I do not see any mention of a need of, or offering of a special puller for the Greer spike. The only thing mentioned that is different about the puller is the way it is applied, which is parallel to the rail rather than perpendicular to it, as is the usual case. The Greer spike requires that the puller be placed parallel to the rail because there is no feature to grab on the spike if the puller is placed perpendicular to the rail.
Beyond the fact that the Greer spike requires the puller to be placed parallel to the rail, Greer claims an advantage for doing so, in that the puller fulcrum is placed on the rail base rather than on the tie. The advantage of this is that the rail base is always solidly available, whereas the tie surface may be rotted or chewed up, and thus require the use of a shim to create a solid support for the puller fulcrum.
However, for reasons that we have discussed about the rail base being a poor support for the puller fulcrum due to its taper, and the possibility of smashing your hand between the puller and the rail, I suspect that the RGS developed its own unique spike puller to avoid the problems just mentioned. That would be the “F” shaped puller. This configuration would permit the puller to be placed perpendicular to the rail, as is ordinary practice, thus avoiding the abovementioned problems with placing the puller parallel with the rail. I can see where actual users of the spike would have preferred to avoid the abovementioned problems more than they would have preferred to take advantage of Greer’s claim of not needing to use shims in case a tie surface was unsuitable.
The ideal “F” shaped puller would deviate slightly from the letter “F” by having the lower half of its stem moved to the right so it meets the lower cross bar at its center. This would avoid any twisting motion arising from a lever offset that would be inherent if the puller was shaped exactly like an “F”.