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I just had a registered user modify a link on my site as a test. The modification was a seccess (after approval and cache clear), but the Modify page is slightly confusing to a registered user.
When a registered user clicks on the modify link they are taken to a page with a Password field and a Send Password field. These two fields serve no purpose for registered users, and are incomplete for non logged in users (no Username field).
The registered user needs to scroll on down and click on the "Modify" button on the bottom of the page, (no other action is needed on this page) and the user will be taken to a page where the actual modifications can be entered. They can make and save their changes from here.
Afterwards all that is required is Admin approval.
HTH.