the CORRECT oil is some engineers opinion, they specify all the aforementioned oils with the same result, the transmission works. although I agree that some form of gear specific oil should be used as there is extreme pressure additives to deal with the high shear loads encountered in a gear box. However that doesn't change the fact some engineer has specified any of the above oils at one time or another.
But if a transmission never specified ATF, for example, as the correct fluid, filling with ATF can/will cause premature failure. The engineers don't make specifications based on "opinions", they do lots of testing to determine what works best. As far as using standard or synthetic, that can even come in to play, as many later model transmission call for synthetic only (like the NV4500), and use of regular gear oil will ruin them.
However, with a trans that originally specified a conventional oil that has a synthetic equivalent, then there's no issue with using the synthetic, other than making sure the seals and gaskets are in good order to prevent leakage.
