docs: add local test running guide to AGENTS.md (#7442)

* docs: add detailed local test running guide to AGENTS.md

Expand the Testing & Validation section with step-by-step instructions
for running backend (Mocha), frontend E2E (Playwright), and admin panel
tests locally, including prerequisites and single-file examples.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

* docs: add Playwright browser prerequisite check to AGENTS.md

Agents and developers should verify Playwright browsers and system
dependencies are installed before running frontend/admin tests to
avoid silent failures and timeouts (especially webkit).

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

* docs: fix plugin test path and browser list per review

- Plugin tests are at repo-root node_modules/, not src/node_modules/
- Playwright runs chromium and firefox only (webkit is disabled)

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

* docs: fix install-deps working directory to run from src/

Playwright is a devDependency of the src workspace, so install-deps
must also run from src/ to match the correct Playwright version.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

---------

Co-authored-by: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

AGENTS.MD

@@ -55,13 +55,89 @@ pnpm --filter ep_etherpad-lite run prod         # Start production server
 
 ### Testing & Validation
 - **Requirement:** Every bug fix MUST include a regression test in the same commit.
-- **Backend Tests:** `pnpm --filter ep_etherpad-lite run test`
-- **Frontend E2E Tests:** `pnpm --filter ep_etherpad-lite run test-ui`
-- **Admin Tests:** `pnpm --filter ep_etherpad-lite run test-admin`
+- **Always run tests locally before pushing to CI.**
 - **Linting:** `pnpm run lint`
 - **Type Check:** `pnpm --filter ep_etherpad-lite run ts-check`
 - **Build:** `pnpm run build:etherpad` before production deployment
 
+#### Running Backend Tests Locally
+
+Backend tests use Mocha with tsx and run against a real server instance (started automatically by the test harness). No separate server process is needed.
+
+```bash
+# Run ALL backend tests (includes plugin tests)
+pnpm --filter ep_etherpad-lite run test
+
+# Run only utility tests (faster, ~5s timeout)
+pnpm --filter ep_etherpad-lite run test-utils
+
+# Run a single test file directly
+cd src && cross-env NODE_ENV=production npx mocha --import=tsx --timeout 120000 tests/backend/specs/YOUR_TEST.ts
+
+# Run unit tests (Vitest)
+cd src && npx vitest
+```
+
+- Tests run with `NODE_ENV=production`.
+- Default timeout is 120 seconds per test.
+- Test files live in `src/tests/backend/specs/`.
+- Plugin backend tests live in `node_modules/ep_*/static/tests/backend/specs/` (at repo root) and are included automatically by the test script.
+
+#### Running Frontend E2E Tests Locally
+
+Frontend tests use Playwright. **You must have a running Etherpad server** before launching them — the Playwright config does not auto-start the server.
+
+**Before running frontend or admin tests, ensure Playwright browsers are installed.** Check and install if needed:
+
+```bash
+# Check which browsers are installed
+cd src && npx playwright install --dry-run
+
+# Install all browsers and their system dependencies (must run from src/)
+cd src && npx playwright install
+cd src && sudo npx playwright install-deps
+```
+
+If `sudo` is unavailable, install system dependencies for webkit manually:
+```bash
+# Check which system libraries are missing for webkit
+ldd ~/.cache/ms-playwright/webkit-*/minibrowser-wpe/MiniBrowser 2>&1 | grep "not found"
+```
+
+If browsers or system dependencies are missing, tests will fail silently or timeout — **always verify browser installation before debugging test failures.**
+
+```bash
+# 1. Start the dev server in a separate terminal
+pnpm --filter ep_etherpad-lite run dev
+
+# 2. Run frontend E2E tests
+pnpm --filter ep_etherpad-lite run test-ui
+
+# 3. Run with interactive Playwright UI (useful for debugging)
+pnpm --filter ep_etherpad-lite run test-ui:ui
+
+# Run a single test file
+cd src && cross-env NODE_ENV=production npx playwright test tests/frontend-new/specs/YOUR_TEST.spec.ts
+```
+
+- Tests expect the server at `localhost:9001`.
+- Test files live in `src/tests/frontend-new/specs/`.
+- Runs against chromium and firefox by default (webkit is disabled).
+- Playwright config is at `src/playwright.config.ts`.
+
+#### Running Admin Panel Tests Locally
+
+```bash
+# Requires a running server and Playwright browsers installed (same as frontend tests)
+pnpm --filter ep_etherpad-lite run test-admin
+
+# Interactive UI mode
+pnpm --filter ep_etherpad-lite run test-admin:ui
+```
+
+- Admin tests run with `--workers 1` (sequential) on chromium and firefox only.
+- Test files live in `src/tests/frontend-new/admin-spec/`.
+
 ### Backend Test Auth
 Tests use JWT authentication, not API keys. Pattern:
 ```typescript