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>

tests/e2e/register.test.ts

@@ -1,6 +1,6 @@
 import {env} from 'node:process';
 import {test, expect} from '@playwright/test';
-import {login, logout} from './utils.ts';
+import {login, logout, apiDeleteUser} from './utils.ts';
 
 test.beforeEach(async ({page}) => {
   await page.goto('/user/sign_up');
@@ -50,10 +50,7 @@ test('register then login', async ({page}) => {
   await login(page, username, password);
 
   // delete via API because of issues related to form-fetch-action
-  const response = await page.request.delete(`/api/v1/admin/users/${username}?purge=true`, {
-    headers: {Authorization: `Basic ${btoa(`${env.GITEA_TEST_E2E_USER}:${env.GITEA_TEST_E2E_PASSWORD}`)}`},
-  });
-  expect(response.ok()).toBeTruthy();
+  await apiDeleteUser(page.request, username);
 });
 
 test('register with existing username shows error', async ({page}) => {