I think there's a subtle difference between what you and the OP have done.
You saved the folder to an new location ans updated the links in the packaged file, so they now point to the package folder. If you do as the OP says and "copy and paste" to a new location and leave the original behind, I think you'll find the links are still pointing to the same folder where you originally saved the package, not to the pasted location.
If you CUT and paste, or otherwise remove the files from the locations where they were originally saved, ID will find them in the new location.