battery light stays on but running fine

[sambo42xa]

My 2007 (99k) is pretty much doing the same thing. Only differance: car is running at idle and the battery light flutters not to the point it goes out. While running I remove either battery cable or even both and the battery light stays on and does not flutter (car is still running). Put the cable/s back on and the light flutters again.
With car off I removed the battery cable off of the alternator and cleaned the connection since it did have a little bit of corrosion. The pin plug I removed also but did not see any corrosion. Output on alternator was around 13.8 and the same at the battery. No change after cleaning connection.
From what I have read here, it seems you folks are steering more towards the alternator (internals) than the battery. No problems starting car so I'm pretty sure the battery is fine (13.8 volts). Everything else works fine. Are there any readings on the pins I can check or are we 99.9% sure it is the alternator?
I'm good with mechanics and I'll check/test whatever.
Thnx in advance!
sambo