Aha, it's the blurb for an image in the carousel for the article and not in the body text.

So the good news is you aren't seeing things, that is what the Ars article said. The bad news is that Ars are wrong and these are clearly not private keys but shared secrets.

Other people have linked Microsoft's documentation, the control panel for Cosmos DB lets you ask for new keys and warns you they may take minutes to be ready, so that means it's a shared secret, as a private key would be something you pick and don't reveal to Microsoft.