Bobbie,
Remember, you do not need to grind any
deeper than the depth of the existing recess, just slightly
larger in diameter that the current diameter of the point-to-point diameter of the nut. Actually, if you look closely at your first photo, at the bottom of the damaged depression, where the debris from the tearout is the worst, the tearout at that point is probably already as wide as you need to use a thin-walled socket, so you only need to widen the rest of the depression to match that clearance.
As for the "thin-walled" socket, that term is superfluous today, as virtually all sockets
sold today are thin-walled, compared to the old ones...and I don't want to hear any cracks about my apparent age because of the "old ones" remark ;-). The cheap ones are thicker because they don't use good steel and would break in use if they were made as thin as the high quality ones. Craftsman, Proto, or any name brand should work.
How thin do sockets get? I only have standard ones, I don't think I've ever seen a thin-walled one. I'll pop by the hardware store and see about grinding wheels and sockets, though at this point I'm not sure I'm willing to try any more grinding.
Remember, you need a VERY TINY grinder bit. The diameter of the grinder bit must be less than the desired width of the clearance needed. If you need a 1/8th inch wide clearance, you should use a 1/16 inch bit or smaller, so you can nibble away at it a little at a time. You can always make it bigger, but you can't make it smaller. Dremel sells such small bits, and probably others as well.
The only other way I can think of would be to use small carving knives to nibble away at the clearance. However, most carving tools are very, very very sharp.
Good Luck!
PS: Ar you going to be at the spring or
fall NOG, or the big Oregon Gathering at Bandon, or the Mary Hill mini gathering, or the WA state gathering? If so, let me know and I'll take a look at it at any of those events.
Mike