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What you describe wanting is an optical sight with correctly set parallax. Parallax can only be correctly set for one range at a time. Rifle scopes usually come either with parallax fixed for 100 yards, or else they are adjustable for different yardages, usually at the front of the scope. You either set it for yardage on a graduated scale or, as you describe, you get the rifle stationary on a target, move your head around and tweak the adjustment until the cross hairs stay put over the target.
I don't recall seeing a pistol dot sight with a parallax adjustment. I think they usually figure your hold and eye location should be repeatable enough for pistol accuracy, but I don't know that should be true. Certainly in practical shooting targets are relatively close and large, so it doesn't really matter a lot. As soon as you get to bullseye matches (conventional pistol) you've got 25 and 50 yard targets, both. I'd want the parallax right for 50 yards in that circumstance.
So, it makes sense to me to make a handgun dot sight with adjustable parallax for conventional pistol matches. I just don't know of one off-hand because I've never thought to look that feature. But at least now you know what it would be called on the specification sheet if you find it: parallax adjustment.
