You set the instructor’s display name that is shown to learners in the course under “Course managers”.
Only verified names can be selected there. To appear, the person must meet two requirements:
They must already have been added to the course as a course manager.
They must have the verified instructor role in OneTutor.
This means that simply inviting the person to the course is not enough. They must also apply for the instructor role and be verified, either during their first login or later via the corresponding link (see How do I get the instructor role?). Only then will they appear in the dropdown for selecting the display name.
For the full guide with all steps, rights and how to invite course managers, see Course Managers: Usage, Rights and Setup.

