The "battery" light indicates that the alternator is not producing any output, so everything electrical in the car is being powered by the battery. It is red to indicate that the car should not be driven when it is on. There is no doubt that the battery will drain down, eventually the engine will stall and that the power steering and brakes will be lost.
A generator works by spinning a coil of wire inside a permanent magnet. An alternator uses an electromagnet instead and the current to drive the electromagnet comes from the alternator itself. An external source of current is needed to power the electromagnet long enough to get the alternator started to the point where it can power its own electromagnet. This is supplied through the small wire attached to the alternator. The "battery" light indicates that the alternator did not start up successfully and is therefore not producing any output at all.
One thing you could try when this happens is to hold the engine at 3-4000 rpms for 5 seconds with the transmission in neutral. Sometimes this can allow the alternator to generate enough current to get the electromagnet started. If that doesn't work, shut the engine off and try restarting it. I wouldn't drive the car vary far with the light on.
My suggestions on the source of the problem are:
1. Loose connection of the small wire on the alternator.
2. Worn alternator brushes not making good contact.
3. An intermittent problem inside the alternator. Testing won't show the intermittent problem unless it happens to act up while the test is being performed.
