Add e2e tests for server push events (#36879)

Add e2e tests for the three server push features:
- **Notification count**: verifies badge appears when another user
creates an issue
- **Stopwatch**: verifies stopwatch element is rendered when a stopwatch
is active
- **Logout propagation**: verifies logout in one tab triggers redirect
in another

Tests are transport-agnostic in preparation for a future WebSocket
migration.

---------

Co-authored-by: Claude (Opus 4.6) <noreply@anthropic.com>
Co-authored-by: wxiaoguang <wxiaoguang@gmail.com>

tools/test-e2e.sh

@@ -34,6 +34,9 @@ INSTALL_LOCK = true
 [service]
 ENABLE_CAPTCHA = false
 
+[ui.notification]
+EVENT_SOURCE_UPDATE_TIME = 500ms
+
 [log]
 MODE = console
 LEVEL = Warn