Stop from Samsung Galaxy S4 turning on when receiving text message
It looks like having the screen turn on when SMS messages arrive is a stock feature of the Samsung Galaxy S4 on Verzion (possibly applies to other models and carriers as well).  I saw threads here and here discussing the problem.  There is no setting to disable only the screen turning on.

The workaround that I ended up using is as follows:

1) Disable notification in the stock app: go to settings and scroll down to Notification settings and unselect "Notifications".



2) Install SMS Popup.

3) Enable SMS Popup.  Then go to Addition Settings and disable "Screen on for Popup".  



Maybe someday Samsung/Verizon will fix the problem, but for now this workaround seems to work for me.