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It doesn't look like any of the contacts are missing but seems like the plating on a couple contacts has started to wear off. So long as the actual pad itself is not lifted or eaten away it is possible to replate the contacts, using an electro plating process. I've even seen people put a very thin coat of solder over a contact to keep it from corroding, though that isn't something I would recommend. It might be cheaper and less time consuming to just purchase a new copy than replating your current copy if you do not already have the tools and materials. You could even order replacement PCBs and swap over all the components. I found a listing on PCBWay.com specifically for pokemon GBA games listed as AGB-E02-20 | Game Boy Advance | Replacement PCB
Sounds like you need to find another source instead of aliexpress or ebay. Digikey has them with the correct 2 wire connection. It's just a standard 18650 battery with the wires spot welded to the positive and negative terminals. Anything with a 3rd wire is likely meant as a charge sensing wire for a batter that has on board circuitry.