13 volts is too low. Charging system voltage must be between 13.75 and 14.75 volts. Yours is only slightly higher than the 12.6 volts of a good, fully-charged battery which indicates the system is not totally dead. The suspects for low output are a slipping drive belt, a shorted cell in the battery, and most commonly, one failed diode of the six inside the generator.
The test that needs to be performed is the standard "full-load output current test", and that requires a professional load tester. The two optional generator sizes for your car are an 80-amp and a 90-amp unit. That is very close to what you should find during the full-load output current test. If there is one failed diode, all you'll be able to get is exactly one third of that, or less than 30 amps. That's not enough to meet the demands of the entire electrical system under all conditions. The output is three phase current which is very stable. When a diode has failed, one of those phases is missing. It's the resulting momentary drop in output voltage, and therefore current, that is detected by the voltage regulator which turns on the "Battery" warning light.
Have the charging system tested completely, then post the numbers here and I'll analyze them for you and figure out where to go next.
Here's a link to an article that may shed more light on the topic:
Tuesday, March 5th, 2019 AT 7:55 PM