If you generate a document without author line, the document properties of the generated PDF show:

In English the field is called "PDF Producer" which is more ambiguous in what it could mean than the german "PDF created with" literally translated.

This is imho the correct thing to have in this field,
which is also backed by the example at the end of section 14.3.3 in https://opensource.adobe.com/dc-acrobat-sdk-docs/pdfstandards/PDF32000_2008.pdf which shows:

But if you set an author line on the document, that line is used as Producer:

In English where the field is just called "Producer" both values could fit, though it is a bit inconsistent.
But the example in the standard and the localization of Acrobat Reader suggest that the author should not be put there.