I didn't read through any of the other replies so this might be repetitive, but ...
Properties will vary from scope to scope as the internals (how they're arranged and spaced apart) will dictate performance more than any exterior feature (tube dia, obj. dia, etc.), so a blanket statement that this or that feature is better for this or that on ALL optics is rarely ever true. This is why some scopes from a given brand/model line are great and others from the same brand/model line aren't.
The only thing a larger objective lens will get you is better light gathering capability in low-light situations and a slightly larger exit pupil. Under "normal" shooting conditions the human eye won't be able to tell any discernible difference between a 50mm and 56mm.
Eye strain is more a factor of lens clarity, resolution and contrast, and assuming both the 50 and 56mm use the same quality of lenses SWFA is correct in stating that one will be no better than the other in that regard.
As far as being less forgiving, the scope shadow you speak of is typically caused by poor eye alignment (to high/low, left or right, or to close/far from the eye piece). Given the 56mm would need to be mounted higher, and as such would affect one's ability to get a proper and consistent cheek weld on a stock with a non-adjustable comb, I would tend to agree with SWFA's claim that it would be less forgiving.
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