Not sure with Bally, but it is common in the electronics industry to have a part number for the bare circuit board with no parts installed. Circuit boards are sometimes contracted to an outside supplier that specializes in building bare circuit boards, so they would need a part number for the unpopulated board. Then a Bally factory may add another part number after the electronic parts are added to the bare board. And there can be a 3rd number, called an assembly number, after final parts such as specific eproms, plastic pull handles, etc for a specific customer or machine are added. You can use the same bare circuit board for different purposes by leaving off some unneeded circuit parts, or by adding a mod such as a piggy-back board. This would result in several different high level part numbers, one for each variation, but the bare board number would be the same since all the various final product boards started with the same bare board.
There is a reason for all of them, but with Bally I think the "AS" number is the one most often used. As an example, AS-2983-201, I don't for a fact but always figured the "AS" meant "Assembly", the next 4 digits (2983) meant the board number and the final digits (201) meant the specific version of that 2983 board.