Notifications ignore your good intentions. Mute Discord, mute the OS, then close anything that can still badge or sound.
Discord
Streamer Mode → disable notifications and sounds. Full walkthrough: hide Discord notifications on stream.
Windows
Settings → System → Focus → Alarms only (or Focus assist → Alarms only on older builds).
Mac
Control Center → Focus → Do Not Disturb. Share across devices if your phone can mirror banners.
Closing still wins
DND hides banners; closed apps cannot notify. Use close all apps before going live.