feat: sync notification badge + clarify cloud storage support

Notification system:
- When sync applies data (file folder, browser account, or sync code),
  ss_sync_notification is written to local storage with the source.
- Service worker catches it via storage.onChanged, shows a purple 'SYN'
  badge on the extension icon that persists until Options is opened, and
  fires a desktop notification ('Settings updated via sync folder — open
  Options to review').
- Clicking the desktop notification opens the Options page directly.
- On service worker wake, SYN badge is restored if the notification was
  not yet dismissed.
- Opening Options clears ss_sync_notification, resets the badge, and
  sends a sync:notification-seen message to the service worker.
- Added 'notifications' permission to both manifests.

Cloud storage clarity:
- Options page now explicitly lists that the folder sync works with any
  cloud storage that has a desktop sync client: Dropbox, OneDrive, Google
  Drive, iCloud Drive, Box, pCloud, Nextcloud, Synology Drive, etc.

https://claude.ai/code/session_01TKpSR9M8JgHLXCp5CeDsQP

src/lib/sync.js

@@ -65,7 +65,7 @@ const SilentSendSync = {
         }
       }
 
-      await this._applyData(data);
+      await this._applyData(data, 'code');
       return { success: true, importTime: new Date(data.lastModified).toLocaleString() };
     } catch (e) {
       return { success: false, reason: e.message };
@@ -131,7 +131,7 @@ const SilentSendSync = {
       const json = chunkKeys.map(k => chunkResult[k] || '').join('');
       const data = JSON.parse(json);
 
-      await this._applyData(data);
+      await this._applyData(data, 'browser-sync');
       return { imported: true, time: new Date(data.lastModified).toLocaleString() };
     } catch (e) {
       console.warn('[Silent Send] pullFromSyncStorage failed:', e);
@@ -154,8 +154,12 @@ const SilentSendSync = {
     };
   },
 
-  async _applyData(data) {
-    const toSet = { ss_lastModified: data.lastModified };
+  async _applyData(data, source = 'unknown') {
+    const toSet = {
+      ss_lastModified: data.lastModified,
+      // Signal the service worker to show a badge/notification
+      ss_sync_notification: { source, time: Date.now() },
+    };
     if (data.identity !== undefined) toSet.ss_identity = data.identity;
     if (data.mappings !== undefined) toSet.ss_mappings = data.mappings;
     if (data.settings !== undefined) toSet.ss_settings = data.settings;