I'd like to know if the Google Tasks API provides the option to move a task to a different list or if I need to create a copy of the task.
The move method seems to only allow moving within the same list.
The tasks API does not support moving tasks from one list to another, just changing a tasks parent task or moving it to the top of the list.
You can however get the task with /lists/tasklist_id/tasks/task_id
, check each of its fields and set populate the same fields with a different tasklist_id using /lists/new_tasklist_id/tasks
and then delete the old one with /lists/tasklist_id/tasks/task_id
Google also provides the Tasks API explorer which should help you experiment.