A model number may be useful if you have it handy or better yet, a few photos. I've done a little bit of research on my end and this may be of use. It is somewhat long winded, bear with me.
I have a Blue Point (Snap-On) seat grinding set with Blue Point drivers (very similar to a Black and Decker I used to have).
I have one Black & Decker driver that is of a different thread and size than my other drivers (they are 11/16"-16 thread). I have not measured it, but none of the stones and adapters I have will fit it. It has a single stone on it about an inch in diameter. The part number (50638) comes back as a 15° stone. As tst suggests, this is a K.O. Lee rebrand, but in the book I found online, it shows it as a 9/16"-16, not a 5/8"-16 as your post states. Goodson does have the adapter for 9/16"-16. I can't find anything anywhere about a 5/8"-16 driver thread. Might be cheaper to find different drivers.
I was looking on eBay and found a
listing for a K.O. Lee seat grinder set. Very interesting looking set. I have a few pieces that would have been for this set.
Vintage Machinery Black & Decker Automotive Catalog This catalog is somewhat informative.
I've attached two photos of what I have. The stone has a Black & Decker part number on it. It also has the rounded six-point drive ball socket on the other end.
I hope this is somewhat useful.