The title of their blog post is: "Linode Manager Security Incident" and that's exactly the name of the customer website where you can manage your instance, billing, etc.
I think someone found a way to gain access to any Linode customer account through the customer website and from there shut down the instance, changed root password and rebooted (you can do that from there).
I think it's just poorly worded - from the OP and what I've read elsewhere someone accessed the portal used for customer service employees that has access to options/all hosts.
Yes the notes in the original pastebin post kind of indicated it was an issue with a "customer support" control panel, maybe it's just another method or area using the Linode Manager.
I just wish they posted a little more, it feels vague.
I think someone found a way to gain access to any Linode customer account through the customer website and from there shut down the instance, changed root password and rebooted (you can do that from there).