I am concluding that the "alternator" light came on and not the "battery" light.

So, the vehicle is trying to tell you that the alternator is not charging.

If you bring the battery to Advance Auto, they will tell you that it is discharged and that it needs a recharge before they can load test it.

With a properly recharged battery, you need to follow the advice above and check to see if you have about 13.5 volts with the engine running above 2000 rpm.

My guess is that the alternator was rebuilt "on the cheap" and that it has no output. You will get about 12.0 volts with the engine running.