The alternator is good the battery is good if just in case one of the three wires is messed up or whatever reason the charging system battery light is on will we be fine or left stranded somewhere.

If the battery light is lit, the diagnosis needs to continue. inspect clean, tighten or replace, connections as required. attach a voltmeter to the battery what is the reading? start the engine, let it warm up, what is the reading? What are the fuses condition, check the fuse panel, and the power distribution box. if you see a charging system fuse, what is its rating ? is it still a good fuse? there has been reports you may need to increase that fuses rating and replace it. When it comes to connections, you may need correct , clean or improve hardware to increase surface area contact. both ends of both cables. what is your volt reading, engine running, engine warm? You can charge up your battery and take the car for testing to verify your conditions and run tests to find which of several tests it may fail. Generally, you can trust the parts store findings, but a technician can run all the tests, find the cause, etc.

The first indication you could expect from that light being lit after the engine runs, is the car is using up the battery and not being charged. Do not remove connections with the engine running, do not effect repairs with the battery connected. you can change the fuse under any circumstances engine off.