I run my iPhone with nearly all sound and lock screen notifications turned off.
That's right. For every app I go into the appropriate settings and turn off anything that produces a notification. I have enabled a handful of apps to display items in my lock screen. I even go so far as to disable the "badges", the little red number that appears at the corner of an icon indicating there's stuff to look at. This has greatly quieted my phone.
Also, unless you are my wife or daughter, your text won't alert me. I turned off all text notifications, so I'll only notice your message when I look at the phone. (I do look at it regularly enough.) If you really needed to convey an important message to me, you can call me.
Of course the one thing I can't turn off are work notifications, but I try to set limits there too. For Slack messages, my phone makes a noise, but I have disabled nearly all the social channels and some of the direct messages as well. I also make use of the "active hours" feature. After a certain hour, I'm not reachable via Slack. If you really needed to reach me, you can raise a PagerDuty incident to my cell.
I try to remind myself that before smart phones and flip phones, we were pretty much not reachable unless you were near a phone, and it had to be a landline phone at that. Instead of attending to every beep and buzz of our smart phones, we used to have to walk with our head forward, looking in front of us and around us.
All this is in-line with my low-information diet, but it's also a way to control distractions. Turning off the notifications helps me remain somewhat present. I have noticed people making good use of the iPhone's "Focus" mode. This is probably something I should explore but my current system works for me.