It will not Edit/Update the Delivery.
The PUT Command Replace the existing Delivery Group information for the Name or ID provided in the request.
I think some clarification in the documentation to explain this should be very helpful.
I notice that as a Bug because the Title of the chapter stand for Edit and in fact it’s not really editing the configuration (my own opinion).
However, I tried to work on a PSH script that will effectively permit to Add information to a Delivery Group, for now, I focus on adding Device Policies.
The script should be released soon, stay tuned …