When using oEmbed on Discord, make sure you include your website domain in the discovery URL. In my <head> element, I specified my oEmbed support like:
<link type="application/json+oembed" href="/path/to/oembed.json">But Discord (truth be told, they're all I checked) wasn't including it in their embeds until I changed it to:
<link type="application/json+oembed" href="https://www.example.com/path/to/oembed.json">So maybe it's not a bad practice to include the site domain in all href attributes.