feat: migrate e2e test infrastructure to @playwright/test

Replace direct playwright dependency with @playwright/test and add
playwright.config.js for centralized test configuration. Update run.sh
to invoke 'npx playwright test' instead of running test files with node.

tests/e2e/playwright.config.js

@@ -0,0 +1,16 @@
+// @ts-check
+const { defineConfig } = require('@playwright/test');
+
+module.exports = defineConfig({
+  testDir: '.',
+  testMatch: '*.spec.js',
+  timeout: 30_000,
+  retries: 0,
+  workers: 1,
+  reporter: [['list']],
+  use: {
+    baseURL: process.env.TARGET_URL || 'http://localhost:8099',
+    browserName: 'chromium',
+    headless: process.env.E2E_HEADLESS !== '0',
+  },
+});