adding a charging fault light

You guys are talking about several different types of alternators all of which are wired differently. The Denso alternators have a warning light circuit that is not part of the excitation circuit. The alternators that use a light or resistor for excitation have a warning light or field circuit for excitation. The sense circuit is not used for excitation or warning it is used for the alternator to monitor battery voltage.
On the old Mopar alternators, there is no circuit for a warning light. If you are an electronic hobbyist, or know one, There is a real simple circuit that can be made using a 741 Op Amp as a voltage comparator that will turn on an LED at a preset voltage. You can set it to turn on the light at any voltage you want. I used to have one set for 12 volts. If your battery is at 12 volts, your alternator is dead. Sorry, I can't find my schematic, but if you Google 741 Op Amp you'll find plenty of them. If you've ever built an electronic project, you can build one of these.