Thanks for all the tips, guys. I normally never stir coffee--when using a bit of cream I add it first, then let the Keurig brew into it--but when a wisdom tooth was removed this summer, I had to rinse with warm salt water several times a day for a week, so stirring-to-dissolve was involved. Consequently, several of my Buffalo china-style diner mugs have exactly the grey marks in the bottom described above.
Had hoped my brief dalliance Bubble Bandit dishwasher detergent would take care of them, but no. It did remove coffee stains from the interior of the mugs, though.
It's about time I actually take care of the issue rather than gazing into a mug every morning thinking, "Gee, that's ugly." (Aside: Having spent 40 years of my life scheduled down to the minute, I've adopted an exceedingly laissez faire attitude in retirement that some might more accurately call rudderless drifting, lol.)
I don't have-and should pick up--Bar Keeper's Friend; but I do have baking soda and a couple of those Magic Eraser sponges.