With rise to wake disabled, notifications cause screen to turn on when picking up phone.

I have rise to wake disabled and when getting notifications I can usually feel the vibration. Upon getting my phone out of my pocket my screen turns on (to I assume show me the notification, even though I have face ID and rise to wake turned off). Is there any way to turn this feature off? I don't want the screen to turn on ever when I move it. I'm okay with notifications, I like those. Just not my screen turning on without my approval basically.