Before I had an HTC one with 4.x & 5.x with custom roms. Whenever I got a text message and my phone had the screen off, the screen would turn on. I didn't think this was special behavior.
I just bought a Moto G (2nd Gen) with 5.x but when I get a text message the screen stays off.
I downloaded an app called AcDisplay and this helps, but it is a suboptimal solution since it shows its own screen and not the default android lock screen.
Is the turn on screen during notification only on custom roms?
This is hard to search for since nearly all results are for adjusting the privacy of the notifications that show on the lock screen.