Here is where the confusion comes from:
[in List settings ContentTypes:]
In the (List) Task ContentType the "Task Status" internal name is Status
[select the source ContentType Task:]
In the (Site) Task ContentType the "Task Status" internal name is TaskStatus
Same internal field renaming (but why?) with:
- DueDate (List ContentType) => TaskDueDate (Site Column)
So when adding the DueDate Site Column to another List or Document Library, TaskDueDate has to be used as the CSR Internal name (and DueDate for the 'same' column in the standard Tasks List)