Alternator or battery???

If you don't run it at FAST idle, the charging system is running too slow to put much charge into the battery. So the test you ran is not an adequate test by any means; you have not proven/tested anything new yet; you know what you knew before.... that something is not right.

Follow the instructions given to get usable test results:
- Clean the terminals and all heavy wire connections everywhere very thoroughly, and the voltage regulator connection to chassis
- Charge the battery overnight; then remove the charger and let the battery 'rest' unconnected for a couple of hours to dissipate the surface charge and then measure the battery voltage. If it is charged and holding, it should read 12.6-12.7 volts.
- Load test the battery at some point in time to be complete; you have a heavy engine cranking load to support
- After the battery is properly charged, then run the car 'til is is warm and at FAST idle, check the voltage across the battery and see if it falls in the range of 13.7-14.3. If so, then you are good on the battery and charging system. If not, THEN it is time to look at the charging system further.

Or, throw parts at it if that is your preference. Some folks like that; OK by me..... but, again, you don't have any usable test results yet. Ultimately, it may be the charging system but you don't know what part, or if there are other underlying problems that will prevent ANY alternator from working right (like bad grounds or other connections...)