There are 2 ground wires coming off of an 02's battery: a bigger one, and a tiny one. The tiny one breaks loose or gets very worn right at the connection to the car. PS: if that tiny ground wire breaks totally, your car wont even run!
Also, the first gens have the infamous positive wire melting problem. Pull the plastic loom off of your 2 positive wires. One goes to the starter, one goes to the fuse box. Check the whole run of each for a melted spot sitting against metal.
Coil pack can cause a batt light, but you would be experiencing bad surges/ power cut-outs if that was it.
I recently put a cheap autozone acc belt (they didn't have Gates) on my zx that must have been an inch too loose. Sqealed like a pig on start-up and under heavy bass music load on the alt. It would constantly give me a batt light cause the belt wasn't turning it well enough. I went and bought a gates after a few days of it, and all back to normal. In your case it could be the tensioner or idler pulley failing and slowing the belt. Do you hear any metal scraping sound from the pulleys when running?
Oh! 1 more!: Another time I got a batt light I found that my positive wire at the connection to the alt was broke and screwed up. Coulda burnt my car down it was so bad, so def check that just to have additional piece of mind. And while you're there, check the harness connection to the alt (harness with 3? little wires) for a broken wire at the harness.
If all else fails, run the car while you pull one fuse at a time til the light suddenly goes away. That will reveal a short. I would start with removing the cig lighter fuse. Those sockets for the lighter short out easy.
Sorry for the long read. Trying to throw a few ideas your way. Hope it's an easy, cheap fix for ya!
I was in your boat the other day sorta. Bought a zx3, and as soon as the delivery driver drove away, the dang car wouldn't crank. Had plenty of juice but wouldn't try to crank. After a few minutes I remembered the clutch safety switch problem they get. I stuck a paper clip in the clutch harness, boom.... starts like a champ now. Free fix... ?