Troubleshooting of Battery vs Alternator

it’s been a while! Hope everyone is well.

This might be kind of a dumb question but here we go:

My car’s battery light turned on a few days ago, then suddenly went off one day when I rode the car to a close neighborhood. I figured it was fine and managed to drive there, turn off the car, turn it on and drive the way back with no issues. Next morning I tried to turn on the car and it would light the dashboard but not enough charge to turn the starter. I took the battery for a charge on a friend’s mechanic shop and mounted it back to make a small voltage test. I checked the battery voltage (~12.8V) then turned the car on and checked again, noticing it actually dropped (12.2V). I figured it was the alternator, so I quickly turned the car off and disconnected the battery so I could keep most of its charge for a quick ride to the mechanics when I had time.

A few days later, I reconnected the battery only to be surprised that it couldn’t start the car again! After this last turn of events, with the battery possibly discharging disconnected or having not been even charged properly on my friend’s place, I’m tending a lot to think the battery is at fault and considering buying one already on the internet (cheaper than local shops).

Am I in the right track to assume that and skipping taking the car to a shop and just getting a new battery?

Some extra info: The battery came with the car already and I own it for about 2 years and some months. It had also discharged maybe a month ago because I forgot a headlight on for an hour, so it could have damaged an already tired battery? (I’m also suspicious that a healthy battery would discharge with only the headlights for 1h)

Thanks a lot in advance!