Nest one content type in another to be able to add single fields I would like to have one content type "Basic" and then another content type "Premium" that has a few extra fields than the basic one. The idea is that I want my users to pay for premium publications (using Commerce Node Checkout Module) if they want to use those extra fields. Basic content will always be free.

It is very likely that I will need to make changes to the basic content type fields that I want to also be reflected in the permium content type. Is there a way to kind of "subtheme" the content type so that it reflects the basic one whenever changes are made but still carries a few extra fields that are independend of these changes?
That's quite a cool question...

I think If I were doing it, I would have the basic form available to all standard users of a standard permissions role...

Then when a customer has paid for the privilege to use more advanced fields, I set up a rule to promote their role to an advanced user role after checkout.

You can then use a module like Field Permissions to determine which fields your basic users can edit, and which fields your advanced users can edit.

And, you only need one content type.