Does the Danish one not contain name etc? The idea was that you'd be comparing that against photo ID.
So no, due to privacy, there no nane show by default.
It's not clear (to me) when you are supposed to show the code with additional information, and I haven't been asked to do so.
The option with more data is only for official situations, like border control. That version is also in English and French.
1. No personal information at all. It only says valid or not valid. 2. Name and date of birth 3. Foreign travel, with name, date of birth as well as information about test type or vaccination type etc.