Rewrite Quickstart guide with type-checked code examples

Rewrites the Quickstart tutorial to provide a more focused, step-by-step
introduction to building MCP Apps. The guide now uses synced code fences
that pull from `examples/quickstart/`, ensuring all code shown in the
documentation is type-checked and tested.

Key improvements:
- Clearer project setup instructions with explicit commands
- Streamlined explanations of the tool + resource registration pattern
- Added screenshot showing the completed app
- E2E test coverage validates the example works end-to-end

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

.prettierignore

@@ -2,5 +2,6 @@ examples/basic-host/**/*.ts
 examples/basic-host/**/*.tsx
 examples/basic-server-*/**/*.ts
 examples/basic-server-*/**/*.tsx
+examples/quickstart/**/*.ts
 **/vendor/**
 SKILL.md