So, if the Trainee is just a renaming of the standard Student role, then that is probably not the issue.
A couple of things:
1) Going back to the screenshot of your settings, you will notice that there are more things there than in the default ones, as in the screenshot that Helen has.
For instance, a Trainer can assign Managers in a course, which is not a good thing. And course creator has all sorts of role assigning power it does not need and should not have. Also, notice that the Authenticated user column, where four roles can assign that. That is not needed and may cause issues.
In other words, there are a lot of odd things on this screen which need to be cleared up first so you can eliminate those as possible sources of the problem.
Basically, setting aside the Enrolment Manager one, those other roles are standard and so this screen should resemble the default screen shown in Helen's screenshot.
In general, it's a best practice not to assign users multiple roles at the site level unless absolutely needed, because the various permissions of the multiple roles can collide and do unexpected things.
That is the first thing I would clear up: regularize that page. You need to simply those settings to eliminate potential role conflict issues.
2) Next, I would test the roles themselves by isolating them:
a) Enrolling a test user account who has NO other site wide role at all except Authenticated user into a course in the AOD Trainer role. If it works, then issue is probably due to the conflict of roles and assignments thing and not to the specific capabilities and permissions of the role.
b) If it does not work, then I next suspect the role capabilities and to test that I would create a new Test Teacher role from scratch based on the Teacher archteype. I would enroll a test user who has NO other site level role at all to be the Test Teacher in a course.
If that solves the issue, then the source of the problem is in the difference between Teacher archetype and the AOD Trainer role you have. (I usually analyze such issues by exporting both roles into their XML files and the comparing the two files in a text editor because staring at those screens is not healthy. However you can also obviously use the Capability overview tool in Site admin > Users> Permissions as well. )
If you have changed nothing but the names, then all those other settings should be the same.
3) Sanity checks
I am sure you have also done a sanity check for your enrollment methods, that they are on and have no weird things like dates.
Another sanity check is that the issue not just that pull down menu. So, I would go into the course as AOD Trainer, and in Users > Enrolment methods > Manual enrolments, click the people icon in the Edit column. This should bring up a left-right two column picking screen. Can you enrol users here? Does the pull down here work? If so, then issue is theme or interface related rather than role definitions. (That would be super obscure, but that's why we do sanity checks ;)
Hope this is helpful.