Thanks bp2008 ! That did the trick. For the sake of helping the visual learners, I've posted some screen shots of the selections you mentioned. These are the options available from right clicking the video within the UI3 web interface and selecting
group settings:

...and the camera labels can be
forced on or
forced off, regardless of the option chosen within the
Blue Iris GUI. If you choose the
no preference option here, then the camera name option within the Blue Iris GUI takes precedence.
Then within the Blue Iris GUI, choosing a specific
camera group and then the
edit layout option gives the user these choices:

...where the camera names (along with many other options) can be toggled on and off. It seems simplest to me to just use the group settings option within the UI3 web interface. I think I have stated the options and the behavior correctly judging from the results that I observed.