As one of the posters suggested, I apparently didn't do a lot of harm to the basic glaze, although I did make the tiles "dull" for sure. I went to a tile shop and they suggested I try putting on a couple of coats of Mop 'n Glo. Got the same advice from a reputable hardware store.
Turns out it did the trick. I can still see places where I rubbed the tiles real hard with the scratch pad, but the overall appearance is shiny, not dull. So I'm out of the doghouse, for now, and bypassed the diamonds.