To define a charge the reading must be higher than the open circuit voltage of the battery, so a good battery will be a good 12.40V, so with the car running ideally you should be seeing 13+ V, under load ideally you don't want the figure to go below 13V.

To answer your question it's your charging system at fault, the battery is fine!

Thats the very purpose of the battery light to indicate no charge, so it would be best to consider a recon alternator, only other thing it may be is the "regulator".

Martin