Thanks for writing back.
In 3.1's control panel, I had "E-mail me whenever a comment is held for moderation"
"Before a comment appears": "An administrator must always approve the comment". [This is in a separate area not related to email notifications.]
So, I did not have any duplicate email item checked.
If I uncheck "email me whenever a comment is held for moderation", then I get ZERO emails!
So something appears to be wrong. Any other ideas?
I also noticed when I posted a comment under a fake name, and when not logged it as the real me, it posted the comment instantly, yet in the control panel it was shown as a comment waiting for approval. (Maybe I should post this as a separate issue).