Charging system issue

Hoping for some help. I have a 2006 Commander, 5.7 l. I have not had any issues with the battery going dead but I frequently am seeing the battery/charging system warning light. It is not on when I start the car but the warning light will illuminate periodically. It will remain lit until I shut down the engine. The next time I start the car, no warning light. Then at some point later the warning light will come on again.
had the battery/charging system check at a local parts store. The battery showed 12 or perhaps a little less with the engine off. Voltage shows pretty much the same with the engine running and alternator functioning.
The warning light seems to come on when there is a bit heavier voltage load. Using the heated seats is one example.
My question; if I recall correctly, shouldn’t the alternator be charging at a slightly higher level? 14 volts is the number I have heard. Could the barely 12 volt output be a sign of a failing alternator? Would that case the warning light. Are there any other reasons the warning light would be coming on?
Thanks in advance for any help or advice.