Is there a single word or phrase to describe a context-less option for "manual" or "automatic"?
Solution 1:
I would go with control.
It gives the right sense of determining who is in charge of decision making: either a human (manual), or a machine (automatic).
Solution 2:
A very general term for this choice would be Mode
Solution 3:
Similar to answers above, I think a general "Automation" or "Automation Level" would be the description for effects gained by one of the choices you mentioned (anywhere from automatic to 'manual', or 'by-hand') and indicate that one was to select at which level one would expect to take or assign control.
Automation:
the technique, method, or system of operating or controlling a process by highly automatic means, as by electronic devices, reducing human intervention to a minimum.
Particularly, as it applies to the technique or method (or extent thereof) of controlling a system in automatic fashion.