Unplug the 3 wire connector on Alternator, this is the wiring for the internal voltage regulator
Turn on the key, engine OFF
Battery light should stay off, if not then light green/red wire has a partial short to ground
If Battery light stays off then most likely issue is the voltage regulator in the alternator is bad
From your description of "running voltage" this is most likely the issue
But test the Yellow wire on the 3 wire connector while its unplugged, should read Battery voltage(12.5v)
This is the Voltage Regulator's "feedback" voltage
".....13.5-13.7 at idle, and 14.5 above 1k rpm or so." <<< this denotes a probelm
After start up battery voltage should go above 14volts, but only for 5 minutes or so, to recharge battery quickly after it was drained by starter motor
Then voltage should be under 14volts, 13.5v to 13.9v the rest of the time engine is running
It can go up when you raise RPMs but should drop back down under 14v within a few seconds, that's the voltage regulator working, the point of having a voltage regulator, its there to maintain a steady voltage regardless of RPMs or AMP draw
So at idle if voltage is 13.7v and you turn on Headlights and Blower fan to HIGH the voltage will drop for a few second but then come back up to 13.7volts because voltage regulator has increased AMP output to match Demand
Same, as said, when you raise RPMs, voltage will go up but will drop back down if you maintain the higher RPMs
Voltage regulator sends 7 to 9 volts to the "rotor"(part that spins in the alternator)
This 7 to 9 volts generates 13.5v to 14.9volts in the Fields(case of alternator) which goes out on B+, stud/nut terminal on back of alternator
That's how it generates the extra voltage/AMPs to run vehicle systems and keep battery charged
A car battery likes to be trickle charged at about 1.1volt higher than its at rest voltage
2 volts higher is fine for Fast recharge , but will "cook" the battery long term
Which is why a voltage regulator is needed
If you raise and lower RPMs and voltage goes up and down with it then there is no "voltage regulation", its just sending the rotor a fixed voltage, say 8volts, and never adjusts it to match RPMs or Demand
That could cause Battery Light to come on, not always
And it can "cook" the battery, shorten its life, if voltage stays over 14volts while driving around at 2,000rpms