I want to display the sender's name rather than their contact photo within a text message conversation
I recommend installing an alternative SMS app. Once you have it downloaded you won't need any Internet connection. From your comment


  "Apps work great when you have a good connection to 4G, 3G, etc." 


you are obviously concerned about Internet speed, but this isn't an issue for an app that sends SMS. It will send and receive your text messages in place of the app you are currently using. I do not mean an Instant Messaging application like Kik or Whatsapp, I mean a dedicated SMS app. The current app you use is provided by whoever made the device (e.g. Samsung or HTC or Sony) and it works in its own way. 

By way of an example here is a shot of Google Messenger  where you can see the name at the top of each conversation. 



Note: This won't be compatible below Android 4.1 JellyBean so will not suffice in this specific instance, but may be of use to others on the site. 

Another good SMS app is Go SMS Pro which is also free on the Play Store. 

Google Hangouts can also be used for SMS. There are many great SMS apps on the store, and I'm sure at least one of them will suffice.