feat(desktop): add macOS dock badge for chats with unseen activity

Show a count of chats (root sessions) with unseen activity on the macOS
dock icon. The count is computed in the existing tray snapshot (full
cross-project list, not the capped tray view; a subtask's unseen rolls up
to its root only when subtask notifications are enabled) and pushed to the
main process over the existing desktop_tray_update IPC, which calls
app.setBadgeCount (0 clears it). The badge clears as sessions are marked
seen on window focus.

Add a Dock badge toggle in Appearance settings (default on, persisted,
darwin desktop only), localized across all dictionaries, with a matching
settings-search entry whose availability mirrors the render guard exactly.

packages/ui/src/stores/useUIStore.ts

@@ -592,6 +592,8 @@ interface UIStore {
   nativeNotificationsEnabled: boolean;
   notificationMode: 'always' | 'hidden-only';
   notifyOnSubtasks: boolean;
+  // Desktop dock badge showing the count of sessions with unseen activity (macOS).
+  dockBadgeEnabled: boolean;
 
   // Event toggles (which events trigger notifications)
   notifyOnCompletion: boolean;
@@ -750,6 +752,7 @@ interface UIStore {
   setNotificationMode: (mode: 'always' | 'hidden-only') => void;
   setShowTerminalQuickKeysOnDesktop: (value: boolean) => void;
   setNotifyOnSubtasks: (value: boolean) => void;
+  setDockBadgeEnabled: (value: boolean) => void;
   setNotifyOnCompletion: (value: boolean) => void;
   setNotifyOnError: (value: boolean) => void;
   setNotifyOnQuestion: (value: boolean) => void;
@@ -877,6 +880,7 @@ export const useUIStore = create<UIStore>()(
         nativeNotificationsEnabled: false,
         notificationMode: 'hidden-only',
         notifyOnSubtasks: true,
+        dockBadgeEnabled: true,
 
         // Event toggles (which events trigger notifications)
         notifyOnCompletion: true,
@@ -1975,6 +1979,10 @@ export const useUIStore = create<UIStore>()(
           set({ notifyOnSubtasks: value });
         },
 
+        setDockBadgeEnabled: (value) => {
+          set({ dockBadgeEnabled: value });
+        },
+
         setNotifyOnCompletion: (value) => { set({ notifyOnCompletion: value }); },
         setNotifyOnError: (value) => { set({ notifyOnError: value }); },
         setNotifyOnQuestion: (value) => { set({ notifyOnQuestion: value }); },
@@ -2243,6 +2251,7 @@ export const useUIStore = create<UIStore>()(
           notificationMode: state.notificationMode,
           showTerminalQuickKeysOnDesktop: state.showTerminalQuickKeysOnDesktop,
           notifyOnSubtasks: state.notifyOnSubtasks,
+          dockBadgeEnabled: state.dockBadgeEnabled,
           notifyOnCompletion: state.notifyOnCompletion,
           notifyOnError: state.notifyOnError,
           notifyOnQuestion: state.notifyOnQuestion,