The voltage regulator is not working properly.
Defective voltage regulator.
Likely the battery or alternator, but probably the alternator
Not likely but as alterantor keeps trying to charge a weak battery it may wear out, however as you are not likely to keep a bad battery in the car for so long there should be no damage to alternator if battery is replaced.
Most likely cable running from stud on top of alternator to battery terminal is corroded or broken.
Not likely.
Battery light usually means alternator not charging. Most common solutions would be be a bad alternator, bad connection, or loose alternator belt. A replacement alternator can be bad. If it is ok than most likely a poor connection to the alternator.
Not likely, but running the engine with a dead battery can damage some alternators. Always fully charge the battery when installing an alternator.
The vehicle should ave a voltage regulator. If the battery and alternator are good then the regulator is most likely bad.
Most likely your alternator is not working properly.
its more than likely the alternator.
If the light is still on then the problem was not the battery but was instead the alternator which is more than likely defective.
More than likely your alternator is getting ready to go out.