Battery engineering has advanced to the point where all you have to do is look at the date code and the warranty.
If your battery has 2 removable covers on the top then remove them and make sure the individual cells are full. Be sure to use proper safety equipment such as gloves and glasses because your working with acid. If not properly filled top them up with distilled water and check to see if that fixes the issue. Sometimes it’s just evaporative loss of the fluid that exposes the plates and prevents them from generating a proper charge. If not then replace it since you already confirmed it’s an issue with the battery. If you are bent on DIY testing on the cheap you could also further confirm this by loading it with the headlights and other accessories to see if there is an excessive voltage drop. Test the new battery then do the same with the old battery for a comparison. Honestly, based on your initial post I would just replace it.
Clean the battery terminals and cable connections thoroughly, removing all the dull gray until shiny metallic gray lead color shows. Then tighten the fittings. I went through this whole thing two or three weeks ago, just as I have seen it several times in the past.
The OP is nitpicking the battery voltage. The difference between 12.7 and 12.5 is meaningless as that can be a variation in the state of charge.
The key is the voltage with the engine cranking (say with fuel pump relay removed) and which should not drop below 10.2 after 10 to 15 seconds of cranking. Hits under 10 the battery is questionable. Down in the 9 or 8 range and it’s bad. That is all assuming a fully charged (or as charged as it will get) battery.