If you want to display a cinemagraph on your website, first add an mp4 video to your MEDIA LIBRARY. We have a guide for video prep and uploading here - Vimeo will not work for this setting. Select the video file and click edit (or double-click the video file). You will see an option to HIDE VIDEO CONTROLS:
In your IMAGES & VIDEO settings, scroll down until you see video autoplay and video loop - activate both settings:
If you want to display THUMBNAIL cinemagraphs, we have a guide for uploading mp4 thumbnails too.