I've seen some folks running around without any ranks being displayed above their character.
Initially I thought it may be linked to the rank insignia on the uniform not being present, but then I saw some have the insignia, yet no rank above their character.

Then I remembered that at some point one could chose the displayed rank via drop-down list, but I don't seem to find it any more. Has that moved to a different location, is it gone completely, or am I just blind and it's in the "Edit record" screen?

I think what you're looking for is the "title" dropdown on the edit record screen. I don't know if you can set it to "none," but it does include all your regular ranks, diplomatic titles, and accolades. Plus "career officer" if you have a lifetime sub. Probably other stuff as well.

It is in the title. When you edit your characters bio in game there is a drop down for you there.

Ah I see.
Only noticed all the other titles in there, didn't notice the ranks in there too.

So, in the end it turns out I'm just blind, thanks for clearing that up