docs: refresh litellm proxy example

[rlippmann]

What changed

• Added an Ollama model entry to the LiteLLM proxy example configuration for local testing
• Revalidated the LiteLLM proxy integration against LiteLLM 1.88.2
• Updated LiteLLM proxy documentation with current validated behavior
• Documented clarification handling as HTTP 400 responses
• Simplified callback import guidance and troubleshooting notes

Why

• The LiteLLM proxy example is a candidate for external visibility and cookbook-style documentation
• The example had not been revalidated against current LiteLLM releases
• Local testing previously required provider credentials; the Ollama configuration provides an easier validation path
• Documentation now reflects observed behavior and current integration guidance

Checklist

• pre-commit run (uv run pre-commit run --all-files)
• tests pass (uv run pytest)