Align LangChain docs with durable SDK recovery

[amavashev]

Summary

• update LangChain and LangGraph guides to the current create_agent and middleware APIs
• document model-, tool-, and request-level gates with durable settlement behavior
• update LangChain JS and Vercel AI SDK examples to current usage and streaming APIs
• align SDK quickstarts, status semantics, versions, cached-token accounting, and failure choreography
• clarify application-facing HTTP 402 translation versus protocol-level HTTP 409 BUDGET_EXCEEDED

The homepage (index.md) is intentionally unchanged.

Verification

• 103 documentation tests passed
• VitePress site, Open Graph assets, and PDF build passed
• current APIs cross-checked against the official LangChain and Vercel AI SDK documentation

Dependencies

Publish after:

• Preserve known spend across framework lifecycles cycles-client-python#95 (runcycles 0.5.3)
• Add durable settlement to LangChain reserve modes langchain-runcycles#49 (langchain-runcycles 0.4.0)
• Never release known spend after execution cycles-client-typescript#180 (@runcycles/cycles-client 0.4.3)
• Preserve known spend after guarded methods cycles-spring-boot-starter#116 (0.3.3)