So if you edit your post using Quick Edit, it won't reload the "Edit by" thing.
I just tested it and it does
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Edit: See.
That's with 1.4; I'm talking about 1.2.
It's a limitation I believe. If you refresh the page, you'll always see it. Besides, if you've edited the post, you obviously know when and who edited it.
This occurs because the page isn't refreshed. Unless the MyBB team add a Javascript feature that refreshes only the edited by, then you'll see it, such as 1.4
Best Regards.