Yes the geographic region you are taling about is about 200 miles inland and again was only known to the Aztec who, did not name the gulf. It is only later referred to as the gulf of mexico as a water rights claim by the spanish. Mexico, the region is superceeded by the Americas naming convention established by europeans as well. As the naming of the gulf of mexico was a political ploy to claim shipping route rights, I still think Gulf of America is more appropriate from a regional standpoint.

As far as Persia and the Persian Gulf or Arabian Gulf, I don't know. These are both equally political claims. To me it seems appropriate to call it the Middle Eastern Gulf. But that would include the Gulf of Oman. Regardless, the names will be whatever the maps say. It's not really up to logic.