docs: add chatbot-backend example showing Vercel Chatbot replacement with Python

[shloimy-wiesel]

Summary

The example/chatbot-backend/ directory now contains a complete, working Python backend that replaces the TypeScript backend of the Vercel Chatbot. This should be documented as a primary showcase of the library's capabilities.

What's included

• FastAPI backend with StreamContext flowing through routes → services → tools
• 6 tools: getWeather, createDocument, updateDocument, editDocument, requestSuggestions (+ chat title generation)
• Redis persistence for chat history and documents (decoupled from the chatbot's Postgres)
• Delta strategy per artifact kind (text=incremental, code/sheet=full content)
• OpenAI-compatible LLM via consume_openai_stream + tool loop
• README.md with setup instructions, architecture, API reference

Verified working features

• ✅ Basic Q&A text streaming
• ✅ Weather tool with full widget rendering
• ✅ Text artifact creation (essays, documents)
• ✅ Code artifact creation (algorithms with syntax highlighting)
• ✅ Document update (full content replacement)
• ✅ Document edit (search-and-replace with replace_all)
• ✅ Request suggestions on text documents
• ✅ Multi-turn tool context (assistant messages persisted to Redis)
• ✅ Chat title generation on first message

Key takeaway

The entire backend is ~600 LOC across 10 files. The library's StreamContext + consume_openai_stream pattern makes the implementation straightforward — no raw SSE string building anywhere.