For future reference: there are fields “URL” (the link) and “Thumbnail URL” (the picture). If you paste the image, the Thumbnail URL field disappears, and picture URL appears in URL field.

Cut it, and Thumbnail URL field appears again. Paste it into the Thumbnail URL field, and enter the link to whatever you want to be opened into the URL field again.

Voila - you have a link AND a custom picture. It’s weird, it shouldn’t work this way, but it works.