Charging issue

I do not have service information for that model, but I used a wiring diagram for an Elantra with a 2.0L engine. This should be similar.

Look at the red wire in the two-wire connector at the generator. That should have 12 volt all the time. If it is missing, check the 20-amp fuse link in the under-hood fuse box.

Does the "battery" light turn on when the ignition switch is turned on? If it does not, check the 10-amp fuse number two inside the car. If it does turn on, the "pre-excitation" resistor is not a concern. The bulb circuit passes the current for initial start-up of the field winding. From then on, a diode trio supplies the current to run the generator. The resistor is only there in case the bulb burns out. I suspect that circuit is working because the generator is doing something, so that means it had to start up.

Thank you for listing the charging voltage. You are right, that is too low. If the voltages going to the generator are okay, that leaves the voltage regulator in the new generator or the belt is slipping over the pulley. A less-common problem that we do run into once in a while is the crankshaft pulley is the cast outer ring on the vibration damper, and the bond can break loose and let that ring slip. Typically you will also have little or no power steering assist.