Found it: ONLY the Assignee or Responsible can change the status
I am both, but don’t know now which was the trigger.

Hello Ronald,

the valid status changes are determined by workflows which can be defined in the administration (Administration > Workflows).
A workflow is defined for a role and a type.
As long as you add a user to your project with a role which allows a workflow transition to status “closed” (e.g. role “Manager”), the user will be able to change the status.
This is independent of the responsible and assignee (even though you can define additional workflow transitions for an assignee of a work package).