@Danny0311 , correct me if I'm wrong, but I think the OP is talking about a situation like this... the phone is sitting next to you on a table, night stand, whatever. The screen is off because it's timed out - it's been sitting there untouched for a while. A notification comes in, but all you hear is maybe the sound. The screen does not turn on to show you the notification. You have to turn the screen on - either with the double-tap, press the lock/power button, or whatever wave gesture or pick-up motion you may have set the phone to respond to.
Unlike the iPhone and the old BlackBerry phones that would turn on the screen to the lock screen for several seconds so you can see from where you got the notification. Either showing the message with it if set to do that, or just that it was a message from an email, text, news app, or a bank transaction. Then, after a set time period, the screen would go dark again.
All the AOD does is keep the time and date showing and a line of icons if you had notifications waiting. By default, Samsung phones did not wake up the lock screen when new notifications came in. But the AOD will add the icon for it when it comes in.
This issue has nothing to do with how quick you get notifications. Generally emails and texts are instant anyway. Bank notifications tend to depend on how quick their system responds. My own bank sometimes notifies me immediately of a transaction. Sometimes a McDonald's card swipe pops up on my phone that instant, while other times it'll take a few hours before I see it.
I'm pretty sure that's the issue. It's something I noticed when I switched from the old BlackBerry Z10 to the S8 back in 2017. That's when I found the Glimpse Notifications app. It was free (still is), did what I wanted, and still use it today on my S20 FE.
I thought there was some setting Samsung added to do the same thing in the One UI 5 update, but maybe this particular feature wasn't really it. I can't seem to find anything in it now to do this without the app.