Why does Honda civic 2003 battery light turn on when I start the car but turns off when revved to over 3000 rpm and turns back off when it goes under 3k?

it means your alternator isn't generating enough juice in the lower revs. Check your alternator belt and see if its loose. If it is loose, then tighten.

I have the battery light too, my belt was loose. I charged my battery.

Check the voltage on the alternator, pumping 18V. Disconnected the battery when running, and the car continues to run. It's not the alternator.

I've started the car and the light won't go away. I've tried disconnecting the battery and the fuses for the ECU and backup, so that the error codes disappear. Light still stays on.

I don't need to drive today, since I already did my shopping.

For me maybe driving after a few clicks the ECU will detect and the light will disappear.