Settings->More->Cellular Networks
Then unclick show mobile network.
At least thats how i turned mine off on the moto x 2013
Posted via the Android Central App
This doesn't exist on the new moto pure. But, the name doesn't stick up in the status bar anyway, only lock screen
Posted via the Android Central App
Lock screen and notification panel.
This is how it's been on all my phones. The phone pulls it from the SIM and displays it. I think you can get rid of it with root but personally I wouldn't root for that.