I'm not sure why they label one 12.7mm and the other .50 MG. They could have just as easily labeled the 12.7mm as .50 pistol. .50 and 12.7mm are the same caliber. For those of you who don't know, caliber is the size of the inside diameter of the firearm. The projectile (bullet) usually has a diameter that is slightly (.001") larger than the inside diameter of the barrel. This is to allow it to stay in contact with the grooves inside the barrel which causes it to spin as it travels down the barrel. The difference between a 12.7mm pistol round and a .50 rifle round is all in the casing (the brass part). A .50 rifle casing is much larger because it needs to contain more powder, in order to propel the bullet further since it is being used in a rifle.
An alternative nomenclature for the .50 BMG is the 12.7x99mm NATO. The .50 AE, which the 12.7mm pistol in game is probably referencing, is also called 12.7x33mm. 99mm and 33mm are the lengths of the cases for the respective rounds.