Definitely the alternator. Replace the brushes and voltage regulator on the alternator and the voltage will go right back up. Note: the Optimum output from your alternator should be between 13.5V to 14.5V.
Have you already fixed it? What was the problem? I have a VW Polo 2001, and I have almost those symptoms... First battery light came up, then later the lights faded and after that I couldn’t even start the engine. Yesterday I recharged the battery and now works, but I think there is an alternator issue for me as well.