Ok, a little more testing.
- Made a copy of my production database (File->Copy from within RM). The hash string for the matches for John H. Peters in the copy was the same as the hash string in the production database.
- Changed the RM Unique ID for John H. Peters in the test database. The same WebHints were there, and the hash string did not change.
- Changed the name from John H. Peters to John Henry Peters without changing anything else. (His name probably actually was John Henry Peters - I just don't have adequate proof). The hash string changed, but the WebHints were the same. MyHeritage did show my RM data as John Henry Peters rather than John H. Peters with the new hash string.
- Changed the name from John Henry Peters back to John H. Peters. The hash string changed again, but not back to the original hash string. And of course the same WebHints were still there. This is most puzzling. I'm going to sleep on it for now. Maybe Tom can figure it out.