Thanks everyone! You made my day!
Checked the reservoir and it was below minimum. My cap says "ATF".
Searching a bit showed that I should use some Dexron III AFT, which I bought at a nearby NAPA. Went for Valvoline, as it was either that or the NAPA branded one. Added it, and noise be gone!
Now I have to figure out where I have a leak and how to fix it. First it will take a couple of days for some of the fluid that spilled when filling to disappear. Then I'll start looking for the leak, but the whole thing is so tight (at least from above) that it doesn't look like it will be very easy.
Once again, thank you for the help. This forum is awesome!
PR