score:15
Thanks for all the comments above. I will answer for my own question. Here is what happened: The first officer was indeed confused about my Canadian greencard and was wrong to redirect me to the E2 Department (it's B5!). After sending another email to B5, I received the following document: the border crossing certificate (Grenzübertrittsbescheinigung (GÜB)) that I can use when exiting the country.
Their response:
Attached you will receive the border crossing certificate to present to the federal police at the airport when you leave the country. Since your visa has already expired, a visa extension is no longer possible.
Note:
This certificate must be handed out to the German Border officials (Bundespolizei) upon exit from the Schengen Area
If this is not possible, you must personally hand it over to a German consulate or embassy outside the Schengen Area. It is not sufficient to send it there by post.
The certificate will then be returned to the issuing authority.