What phone do you have? For my note 3 I can open settings> device> sounds> scroll down to the Samsung applications and there's an option for messages. You can toggle off the vibrate and choose silent for the ring tone but leave on notifications to get the icon on your bar.
It depends on your phone model and texting app you're using, but usually inside the app you can find settings for notifications, and there you can select whether or not you want sound/vibration with your notifications (or disable notifications altogether).
Ok, I'm using the Samsung Note 4.... The first set of instructions aren't exact.... Would I go into the text app itself to change those settings? Please help.