Mazda protege 94 charging system

Mazda protege 94

First, the battery couldn't be charged. I checked the voltage of the battery when the engine is on, only 12.5V. It would gradually increase to 14.1V in 5 minutes.I installed a new alternator. The voltage is only 12.2V. The output of the alternator is the same, 12V.

The belt and tension seem fine. Or if there is belt slippery, how could I know? The battery, though old, could be charged at Autozone.
Mounting bolts are tight.

But I just found out that the horn and/or the horn switch doesn't work. For now, I will just disconnect the horn. Will this affect the charging system so much?

Could anybody help me to find what else could be the problem concerning the charging system? I am brand new here. I feel really frunstrated about the problem. Thanks a lot!

Where did you get this new alternator? I hope you didn't get it at
Autozone. To see the correct voltage when your running you need to check it not at idle. Sometimes the voltage at idle is 12V even with a "good" alternator (although I prefer to see it at about 14). Check the voltage with the engine at 2000RPM.

What's the battery voltage when the engine is not running and everything is turned off, like first thing in the morning. You could also have a bad cell in the battery. I assume that your battery connections are clean?