Show read instead of delivered on my device for messages I send
I'm assuming you are talking about the default Apple messaging app - iMessages. In iMessages, you (as a sender) can set your iMessage Read Receipt to ON/OFF. If it's set to ON on YOUR phone, when a person sends you an iMessage text and you read it, they will get a READ indicator on their phone. 

Conversely, if you send a text to a person who has set their Read Receipt to off, you will only see "Delivered" when the text was delivered. There will be no "read" in this case.

Take a look at this Engadget article which describes this in depth. 


  Apple's iMessage app provides message status updates that let you know
  when a message has been delivered. It also has a handy feature called
  Read Receipts that lets you know when someone has read the message.
  
  This status update can be a blessing when you want someone to know
  automatically that you have viewed their time-sensitive response.
  Apple handles the status update so you don't have to do anything other
  than open the message. Just read the iMessage and the sender will see
  the "Read" timestamp in their timeline.
  
  As useful as the Read Receipts feature may be, it also can cause angst
  both for the sender and recipient. You can imagine the following
  scenario -- a wife discovers she is pregnant and sends an excited
  message to her husband. She sees that he read the message an hour ago
  and is upset because he never responded. On the other end is the
  husband, who viewed the message right before an important meeting. He
  now is agonizing over how to respond while his boss is breathing over
  his back, knowing that his wife is aware of the fact that the message
  has been read.
  
  If you wish to avoid such complicated message relationships, you can
  easily turn off read receipts, which hides the fact that you have read
  a message from the sender. The sender will still see the "delivered"
  status, so they can be assured the message arrive on your handset.
  They just won't know if or when you have read the message. Follow
  these steps to turn off Read Receipts...