I noticed the battery light come on?

Yesterday, I took off one of the leads from the battery to change the airbag module behind the glove box to put back the original module as the one I purchased off of eBay didn't work. After getting everything done, I reattached the battery cable (the red one).

Today while going somewhere, I noticed the battery light come on for the first time since I have had the car (2015). I got to where I was going, which was about 3 miles. I parked and spent around two hours at the location I went to.

When I got back in the car the engine turnover was sluggish as though there was not enough juice to the starter motor, though the engine did start. A friend who was leaving the same location later told me that the lights were dim.

I got to another location (my friend's house) ok, which was about a mile away. The battery light dimmed when the car got up to speed but became brighter when stopped at a light.

I didn't notice anything untoward at this point.

When leaving my friend's house, the start of the car was again sluggish. When getting to the first light near my friend's house I noticed that the engine was losing power so I revved the engine and found that it had picked up power. I had to keep the engine at high revs so that the engine would not stall.

I had to zoom off the line when the lights turned green otherwise the engine would bog down.

There were two more sets of light, and I noticed that if the engine went down to idle (800RPM) the engine would lose all power, so I let the car roll forward and let out the clutch to restart it that way. I then tried keeping the revs high and tried to continue on, but I got another few hundred yards down the road when all power died.

The weather has got cold, but I suspect that either the red lead to the battery was not put on properly, the alternator has gone, or the battery lost charge due to the weather.

The car does have an integrated alarm (a BMW one) with a blinking red light.

In the past I have noticed the blinking red light does drain the battery over time if the car is not used.

But there have been no issues with the battery being drained as the car has been used nearly every day.

Is there a way to determine what has happened without taking the alternator off and getting it tested?
I actually went back to where the car was left, the same night, took the battery out and got it home and charged it overnight. I went to the car this morning, installed the battery, after it had been charged to around 15V (it had 11.9V when I started charging it), and the car started right up but the battery light still showed.

I got home okay, about a mile away, and then took the alternator out and sent it off to be tested and, if necessary, refurbished.

I had the alternator looked at and it is toast! The rotors have completely worn out, so it was the alternator that caused the issue. A new one is being ordered.

Thanks in advance for the help.