Youtube or Vimeo for embedding private video

Hi,

I have a site wherein I want to show a video to registered users.
The video is currently uploaded on YouTube and is unlisted.
SEMrush

I copied the embed codes but it also shows the link “Share” over the video, I don’t want to show that option or any other options to users, since its a video meant for private users.

How do I do that or should I look at Vimeo as an option?