It may not be a bad idea to pull the ashtray and plug in a volt meter to the cigar lighter so you can monitor the charging system from the cabin. I did this for a while, then ended up installing a gauge volt meter on the dash for full time charging system monitoring. (This was after killing 1 battery, replacing the battery, replacing the voltage regulator, VR cracked, overcharged my new battery for a few months until it died, replaced my battery and VR again). I think I'm OK now, but there seems to be a lag time from starting the car to getting a voltage reading much above 12. I like having the gauge though...

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Blue '83 300D "Domobile"
Red '73 Cimatti City Bike (moped = fun)

The open cell (no load) voltage of a 12v car battery should be 12.6v (2.1V per cell) Your measured 12.56v sounds close enough.

Next, you will need to load test the battery. You could take it to Autozone or Advance if you don't have a load tester, or you can get a rough idea by connecting an analog voltmeter across the terminals and starting the car--the voltage should not drop below 10v. If the battery fails the load test, replace it.

Check the alternator belt tension, and make sure that all connections are clean and tight.

Make sure you have the correct light bulb in the charge indicator light in the instrument panel.

Make sure the glow plug relay is shutting off.

Start the car with a voltmeter connected across the battery terminals. In order for the alternator to charge the battery, the voltage must be above the battery voltage. Leave the voltmeter connected and load the alternator by turning on high electrical loads (headlights, heater fan, etc.)
Leave these loads on for a few minutes, and watch the voltmeter. If it starts out ok, but drops too low after a couple minutes, replace the alternator.(you could try just the regulator/brush assy first, but this won't help if you have a bad diode)

Hope this helps

Could be..........it's a low cost attempt at a fix..............