The warning light in a simple system provides excitation current to the alternator until such time as the alternator itself provides excitation current. The light glow indicates current is coming from the battery. The light going out indicates alternator taking over. In a simple engine manufacturer supplied system most critical connections are internal to the alternator if it has an inbuilt regulator. You could check the battery voltage to see if it rises to 13.75 volts when the engine is running (idle) or if it doesn't rise until the engine revs up. This will confirm you have a fault. I doubt a loose belt would give the symptoms. I think the best bet is to remove the alternator for bench check of all the diodes especially the excitation diodes. good luck olewill