My phone

A regular SMS is delivered to those people that are non-iOS users. Those messages would not get a delivered or read notification. Only iMessages will get a read receipt, and for you to see that someone has read your post, they have to have configured their device to send the read receipt. Your turning it on for your phone would send the read receipt to those you are messaging.

The ability to see a Read receipt on your phone when you send someone a message is going to depend on the setting on their device, not yours. They would have had to turn on Read Receipts for you to see Read instead of Delivered, and that also only occurs with iMessage and not a regular SMS. It will also not work for a group iMessage.