In that case, you are not a "consumer" and indeed misunderstood the blog post. This is only available for private individuals, or Verbraucher.
More on that can be read in the law: https://www.gesetze-im-internet.de/bgb/__13.html (§13 BGB).
That said: the support is right.
Thanks for clarifying. Although I have two services, both of which are multi-month terms out of their original contract and was allowed to cancel one with a 31 calenda day notice, but the other referenced in the OP, handled by a different (and very rude) support agent isn't being treated the same for some reason.
In this case, you are not considered a "consumer".
Edit: Well, what kind of service is it? You still have the possibility to offer your product to other users here in the forum. If someone takes it before you get the next invoice, the invoice goes there and not to you.
It's just a RS Black Star VPS, nothing major but was surprised that I am being told one thing for one service and another thing for a similar service with the same contract terms.