2001 Honda Civic alternator or something else?

Ok, this may sound stupid, but I did it to myself.
car - 2001 honda civic Lx 1.7
this saturday morning, I was trying to investigate a strange rattling sound, coming from left side of engine while cold and disappearing after 15min driving. I was'nt sure if it is from ac or alt pulley or from timing belt area. So I took apart alt belt, started the car. Car started fine, sound was still coming from timing belt cover area, so I found out the source of sound. NOW I put the belt back, tightened it and started the car, car again started fine, ran for 2 seconds, then RPMs dropped and came back, then I noticed that battery light was on, on dash board. I knew something wrong with with battery or alternator. I took it to autozone, they checked battery said it was bad. I got a new battery and came back. I replaced the battery but light was still on. I checked new battery voltage it was 12.6, I started the car, voltage went upto 14.4 or 14.5, then turned on headlights, rear defogger, blower, tape one by one. Voltage at bat terminals went down to 12.8, 12.9. I asked someone to rev up the car to 2000 RPMs, voltage came back to 14.2.
then I disconnected the battery while car running, engine went on running, no stopping or stalling.
Finally I checked under hood fuse #9 and under dash 10 & 4. All were ok.
Car now starts and runs fine, with everything working but battery light is on on dashboard. Any suggestion what might be going on.