Just out of curiosity I reinstalled Verizon's messaging app. I don't have the red circle you are referring to, but I believe the repeat names you are describing is where you can customize the color of the message box as well as their background. You did say that you are in the "customize all" section.
If you have the messaging app syncing with the web, I do not, it may also have something to do with that. The app also has a feature that can track your phone with GPS so that you and the recipient, if both using the app, can see where each other are. That circle with the pointer in the middle may represent that the feature is enabled.
If you press the power and volume down button on the back of the phone, this will take a screenshot. Upload that shot here so we can see exactly what you are talking about.