Very nice thread.
Kudos, especially, to Kludge for his expansive and well thought out post.
One of the important differences between shooting a shotgun with both eyes open and a pistol with both open is related to the idea that in the shooter/shotgun system, the shooter's eye is the rear sight. How well the shotgun "fits" the shooter or how well the shooter has adapted to a particular gun determines whether the gun shoots where it is pointed. Thus, a very common, and accurate, comment about shooting a shotgun is that you don't aim a shotgun, you point it. (plus the fact that the targets are moving).
Keeping both eyes open eliminates "sympathetic" dilation of the pupils, in a sense, "stabilizing" both eyes and minimizing muscle tension. If the non-aiming eye is "blacked out", it will dilate open. The other, open, aiming eye is affected by this....it wants to dilate open a bit also. The more open the pupil is, the shorter the depth of field. When aiming a pistol, we want as much depth of field as possible, hence the use and popularity of adjustable apertures like those from Merit and Gehmann.
(A related digression - back in the days of the Cold War, when any type of competition between the USA and the USSR became a battleground for warring philosophies, the Russians invested tremendous time and energy in the shooting sports,especially in "International Pistol" match shooting.
One of the enduring benefits of that time is the publication of A.A. Yur'Yev's "Competitive Shooting". There is extensive discussion of the eye and it's function in the aiming process - chapter 8. it is worth reading. ISBN 0-935998-53-5.)
About aiming and front and rear sights and the targets.....though there are three elements in the system, it is an optical impossibility for the eye to focus on more than one of those at a time. We focus on the front sight as it allows the greatest precision. Notice the big difference between that and shotgun shooting wherein the focus is on the target, sight alignment having been taken care of by mounting the gun.
Pete