The web service functions that would allow you to use Lightwork with the new assignment type are still under development and so Lightwork cannot yet be used with the new assignment module.
Some of these functions are described in the section "The new Lightwork web services" in the development document that you refer to. Note that these functions do not contain the text "lightwork" since they are designed to be generic functions that can be used by anyone, not just Lightwork (your screenshot shows one of these functions - core_enrol_get_enrolled_users_with_capability).
The current version of Lightwork (3.1.X) can only be used with the OLD 2.2 assignment type.
When all the new web services are completed and integrated into Moodle, a new 4.1.X version of Lightwork will be made available that can be used with the new assignment type.