Can't tell if there's an alternator or battery issue

I went to the store the other day and when it was time to leave, my car
wouldn't start. I called a buddy to come to jump my car. He jumped it and the
car started up and I made my way back to my apartment just fine. When I
tried to start my car the next day for work, it wouldn't turn on.

I thought it would be a simple battery issue but the battery I have now is not
even 2 years old (Duralast battery). It's still under warranty at AutoZone. So I
head to AutoZone and the manager tells me that he's going to charge and
test it and to come back later. So, that's what we do. A couple of hours later I
get a phone call that the battery charged to ~90% capacity just fine and (his
words not mine) that there isn't an issue with the battery...So I put the
charged battery in my car, drive around a bit, come back to see that the
capacity of the battery dropped to 66% meaning that the alternator may not
be charging the battery correctly OR that the battery may not be holding a
charge properly.

I was convinced that it was an alternator issue UNTIL a friend and I turned
the car on, disconnected the negative terminal of the battery (while the car is
on), and proceeded to turn on the radio, AC, and high beams. We ran the car
like this for about 20 minutes and it ran fine. So now this leads me to believe
it's a battery issue since the alternator could support all of that for upwards of
20 minutes.

For further information, I tested the voltage of my battery while idle (not
running) and it came out around 12.2 volts. Tested it again (while turned on)
after running the car for about a minute and it came out to about 12.95 volts.
12.95 volts may be a little low while the car is running but I may have not let
it run long enough. I can test that again if anyone thinks it'll help.

Does anyone have any other thoughts? Maybe the AutoZone guy is trying to
prevent me from cashing in on my warranty?

Thanks in advance.