Yes, and if you are a company owner, you usually have a contract between you and yourself? (https://www.hensche.de/Rechtsanwalt_Arbeitsrecht_Handbuch_Ge..., "Geschäftsführeranstellungsvertrag")

So OP has to clarify on both layers: natural person himself, legal entity, and the relation between them. And the result is affected by the questions OP raised?

If you are a shareholder director (“Gesellschaftergeschäftsführer”) you have a contract but are still treated as self employed for most laws. The 183days rule does not apply in that case.