I can consistently reproduce this issue in any browser using the following steps
-Open the Insert Image dialog
-Upload a new image
-Leave the Link URL field blank
-click "Insert Into Post"
-click the image in the post
-click the Edit icon to edit the image settings
-enter a valid URL in the link URL field
Bug: The new link URL is not stored in the post. When I open the edit dialog again for the image, the link URL field is blank again
Expected: the new link URL is stored
I have found that I can get the link URL to update, but only if I add a caption to the image. I have a different blog on the same machine, with all the same settings as this blog AFAIK where this behavior is not happening and editing the link URL works as expected. I even tried a fresh install of WordPress for the problem blog, and that did not fix the issue.
There are no error messages appearing in Firebug or Chrome dev tools. I have pretty much run out of ideas for how to fix this.