Just copy & paste the URL!
Yes, it' that simple! When uploading a video, you can use a YouTube account or Vimeo. Simply copy and paste the entire URL et voila, you’re all set! No need to cut off pieces from the URL any longer.
PS.This won't affect the videos you have uploaded so far, so no need to worry!
Remember: If you want the videos to stay private, make sure you use your own YouTube account and have your personal video settings on “unlisted”.