docs: fix client setup and Responses state guidance

[ryan-lempka]

Fix Codex setup and Rust client examples, and clarify Responses history retention. Docs only.

Summary by CodeRabbit

• Documentation
  • Updated standalone proxy setup instructions to use the dedicated switchyard Codex provider and route, with authentication through OPENROUTER_API_KEY.
  • Refreshed LLM client examples, dependency references, protocol links, and routing usage.
  • Clarified response continuation behavior across native and cross-format providers, including conversation history, routing consistency, capacity limits, unknown IDs, and conflict responses.

[github-actions]

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ℹ️ Recent review info

⚙️ Run configuration

Configuration used: Repository: NVIDIA-NeMo/Switchyard/.coderabbit.yaml

Review profile: CHILL

Plan: Enterprise

Run ID: fcb98e97-a806-4975-b74d-3a2b9e05e90c

📥 Commits

Reviewing files that changed from the base of the PR and between da8133b and 807801a.

📒 Files selected for processing (3)

• README.md
• crates/libsy-llm-client/README.md
• docs/routing_algorithms/llm_classifier_routing.md

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Walkthrough

The pull request updates setup instructions, Rust client examples, protocol references, and Responses continuation documentation. It documents Codex provider configuration, RuntimeModels usage, cross-format history retention, model preservation, and capacity behavior.

Changes

Documentation and Routing

Layer / File(s)	Summary
Codex provider setup README.md	The instructions configure Codex CLI with the switchyard provider and local Responses API. They state that upstream authentication uses OPENROUTER_API_KEY.
LLM client examples crates/libsy-llm-client/README.md	Dependency paths and protocol links are updated. The routing example passes RuntimeModels to run.
Responses continuation behavior docs/routing_algorithms/llm_classifier_routing.md	The documentation distinguishes native Responses storage from locally retained cross-format history. It documents model preservation, capacity handling, and unknown-ID routing.

Priority: ⬇️ Low

Estimated code review effort: 2 (Simple) | ~10 minutes

🚥 Pre-merge checks | ✅ 5

✅ Passed checks (5 passed)

Check name	Status	Explanation
Description Check	✅ Passed	Check skipped - CodeRabbit’s high-level summary is enabled.
Title check	✅ Passed	The title accurately identifies the documentation changes to client setup and Responses state guidance. It is concise and specific enough for the documented changes.
Docstring Coverage	✅ Passed	No functions found in the changed files to evaluate docstring coverage. Skipping docstring coverage check. Docstring coverage is scoped to functions touched by this diff. Analyzed 0 functions across 0…
Linked Issues check	✅ Passed	Check skipped because no linked issues were found for this pull request.
Out of Scope Changes check	✅ Passed	Check skipped because no linked issues were found for this pull request.

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_{A rabbit tunes the route with care
Codex hops through fresh provider air
Runtime models join the run
Stored replies remember which model won
Old IDs stay useful in the queue
New docs make the pathways clear too}

_{Comment @coderabbitai help to get the list of available commands.}