There are two types of voltage regulators - internal and external. The older cars from had external, most newer cars are internal. But we can't leave things alone, some manufactures have moved the voltage regulator to the ecm.
"We checked all the connections, cleaned some, made sure we had good contact, charged the battery to 12.9, sat overnight in minus 10° weather, stated right up this morning, gauge read on the high side of the half mark on the gauge, took it on the highway for about 20 miles, battery light on the whole time".
This is why you need to check voltage output at the alternator post and at the battery post. Do you have the same reading at both?
Field Coil: Not saying the alternator is bad, just saying it might be losing power to the field coil. It is just one of a few possibilities, the other is wiring issues. Could be as simple as the plug is shot.
Need to know the charging voltage at the battery and at the alt output post when the battery light is on.