Use the actual Vimeo video page URL and not the part that is on the provided iframe from Vimeo...

In other words, just use:

I rushed to respond thinking you were inputting the wrong link to the "Media" tab...

What you need to do is actually replace the text fields that you now have for links with textarea fields, so your users can insert the full <iframe> embed provided by either YouTube or Vimeo.

If you wanted to keep the links (so it's easier for users than inserting embed code), you would have to process the output of the extra fields in your item.php template override, so you can transform the link to an actual embed, using for example AllVideos.