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I have a 2002 and my battery light has been on for 6 months and my alternator output is OK just like yours. I've stopped worrying about it. Here is the weird part, when I take a sharp left hand turn up a steep hill about a 1/4 mile from my house and give it a lot of throttle the light will almost always go out - for a couple of minutes. An electical gremlin if there ever was one.
I put a fairly good Midtronics battery tester on a friend's '04 Ford Taurus that shows a Check Battery message or something. It tests the battery and with prompts, guides one through alternator test at idle, raised idle and with a load. The Taurus passed all tests. I found a TSB for the issue and the fix is a PCM update which nobody except a dealer can do of course!
Had this same issue on a Ford Contour that I once owned. If you battery is ok then take some elec parts cleaner or brake parts cleaner and flush the alternator with it. When I did this it solved the problem. Seemed to get full of dust and brush shavings and allowed better contact. I also cleaned all battery and alternator connections where they appeared ok or not.
They did stick the alternator pretty darn low. When I went looking for it, the first thing that came to mind was "wow that's gonna get full of crud and easily get wet in the creek crossings"