I've now got to where I can replicate this problem at will. If I let the engine cool down for 2-3 hours or lift the hood for an hour or so, the Battery Light (Mazda calls it the "Charging System Warning Light"
will light up 10 seconds after the car is started. If I drive it for 10-15 minutes, it then goes out and does not come back on. A few times it would go out and immediately come back on before the 10 minutes were up but it will always go out for good at 10-15 minutes.
I actually replaced the battery with a known good one and the problem was still there so I have ruled out the battery. Tonight, I went through every single connector and wire that I could touch. I disconnected all the connectors and sprayed most of them with DeOxIt connector cleaner, blew them out with compressed air, and reconnected them.
I started the car up and the Warning Light came on. I had checked the voltage on the battery before starting and it was 12.7 volts. With the car at idle and the warning light on, the battery was now 12.2 volts - Not Charging. So I then had my son rev the engine up to 2000-2500 rpm and the voltage stayed at 12.2 volts. This thing is definitely not charging when the light is on.
So I took it out for a drive and after 12 minutes, the light went out. I brought it back home and measured 13.4 volts at idle and 13.9 at 2500 rpm. It was charging now.
Then I shut it off, disconnected the negative terminal, and checked the continuity of the cable running from the generator/alternator to the positive lead of the battery. The cable was fine - 0 ohms.
There is a gray/tan block right at the battery between this generator cable and the terminal. I'm not totally sure what this is. It doesn't show up in any of the on-line service manuals. I guess it could be some sort of protection device - maybe a diode, a fuse, or breaker ?
Does anyone know what this is ???
After the car was warm and the light was out, I measured continuity across this device. I will check it again when it is cold. Otherwise, at this point, the clues seem to be pointing to the generator/alternator and some sort of thermal issue.