I have a Note 9 and also just got a Galaxy watch, which alerts me to my notifications, therefore I dont want my screen to light up whenever I get a notification and I also dont want edge lighting on. I only want my watch to act alert me when I get a notification.
In the Settings menu, scroll down until you see the “Display” entry, then tap that. A little way down this menu, you'll see a toggle for “Ambient Display.” Tap the slider to disable it. That will disable Ambient Display itself, which will prevent the display from waking every time you get a notifications