Remove both the battery and the charger and have them bench tested at an AutoZone or similar -- it's free.
Basically, your battery indicator light will come on whenever the charge circuit is not working. That is why when you first turn your key before you actually start the car, the light is at full brightness -- the engine is not running, thus there is no current being fed by the alternator.
How long have you had the alternator currently on the car? Don't ask me to explain this, because I do not know the details, but sometimes what fails in the alternator is actually the voltage regulator. On our cars, the regulator is integrated into the alternator assembly. Well, I tell you all that to tell you the following: Voltage regulator failures usually aren't apparent until after the regulator eats up, i.e., you will not be able to see it on a bench test at AutoZone. Pay attention to whether the dim battery indicator illuminates as soon as you start the car, or whether it only appaears after you have drive it for a while.
You can always just wire a voltmeter to your battery and keep the voltmeter with you inside the car, checking the voltage. I'd expect to see about 12.8 V (don't quote me on this) with the car off, and about 14.5 V (I am more confident about this figure) while running. If while you are driving, the voltage drops and stays low (~12.5 V or so), then that would indicate that your charging system is ftl.
Does the light come on intermittently? Does it stay on at the same dim level at all times, or does it fluctuate? Does it worsen if you start turning on more electrical systems (radio, A/C, headlights, brake lights, etc.)?
Good luck.
Edit:
I just noticed that you Google up some stuff and found some of the things I covered on my post. Anyhow, I wish you would have Googled first, just out of courtesy. Moreover, I am sorry it took so long to reply. Some of us have jobs at which we actually work during the day.
