Add Logout to system menu and style the SDDM login screen

bin/omarchy-menu

@@ -566,14 +566,15 @@ show_about() {
 }
 
 show_system_menu() {
-  local options="  Lock\n󱄄  Screensaver"
+  local options="󱄄  Screensaver\n  Lock\n󰍃  Logout"
   [[ -f ~/.local/state/omarchy/toggles/suspend-on ]] && options="$options\n󰒲  Suspend"
   omarchy-hibernation-available && options="$options\n󰤁  Hibernate"
   options="$options\n󰜉  Restart\n󰐥  Shutdown"
 
   case $(menu "System" "$options") in
-  *Lock*) omarchy-lock-screen ;;
   *Screensaver*) omarchy-launch-screensaver force ;;
+  *Lock*) omarchy-lock-screen ;;
+  *Logout*) omarchy-system-logout ;;
   *Suspend*) systemctl suspend ;;
   *Hibernate*) systemctl hibernate ;;
   *Restart*) omarchy-cmd-reboot ;;

bin/omarchy-system-logout

@@ -0,0 +1,11 @@
+#!/bin/bash
+
+# Logout command that first closes all application windows (thus giving them a chance to save state),
+# then stops the session, returning to the SDDM login screen.
+
+# Schedule the session stop after closing windows (detached from terminal)
+nohup bash -c "sleep 2 && uwsm stop" >/dev/null 2>&1 &
+
+# Now close all windows
+omarchy-hyprland-window-close-all
+sleep 1 # Allow apps like Chrome to shutdown correctly