SMS Time Stamp (sent, recieved?) (Newbie, bear with me)

I've searched on "timestamp", but couldn't find this issue. The other messages weren't clear (to me).

Iphone (6.1.3, VERIZON). My recieved text messages have timestamp of the time I read (open) the messages. Not the time the sender sent the message.

But, my coworker (AT&T) says his recieved messages are stamped with the time that the sender sent the message. That is what I want also.

Can someone please explain why the difference? If I swap to AT&T will the timestamp change to when sent?

thanks!