Yeah, that's the cross bar.
If the cross bar isn't turning when the key is turned in the handle, that would suggest there's a failure at the back of the key barrel (or the bit that holds the outer end of the cross bar - attached to the exterior handle back-plate).
Might be an idea to remove the door handle and back-plate assembly, although be warned that the two screws holding the back-plate in place are very prone to shearing off when being undone
I've attached a few pics of the new driver's door lock actuator in place (in case it can be of any help to others).
Let us know what you find..
Thanks,
Karl.