The Autozone machine is okay for most checks, but not all.
You have the battery cable to the back of the alternator, then the smaller plug with two wires. The red one is ( or should be ) another battery source. You also have another small wire. This one starts at the ignition switch, and goes to the warning light, and then to the alternator regulator, which is inside the alternator. It feeds an ignition signal into the regulator to let it know it's time to turn on and go to work.
If the light comes on when the key is on engine off, it has a complete circuit, and grounds inside the regulator in the alternator. When the engine spins and the alternator starts to charge it puts power to this line. This creates a hot to both sides of the wire/bulb, and the light goes out due to equal voltage stopping current flow.
Unplug the small wire with the key off. Turn the key on, don't start the engine. If the alt light is on you have a wiring problem. If it is off, plug the connector back in. If the light comes on the circuit is fine. If it stays on when the car is started, you have an alternator/regulator problem.