docs: create client integration guides for OpenAI-compatible and Anthropic-style APIs

[moonming]

Parent issue: #240

Summary

Document the client-facing integration surface for AI agent developers.

Scope

• docs/integration/openai-compatible-api.md
• docs/integration/streaming.md
• docs/integration/tool-calling.md
• docs/integration/anthropic-messages.md
• docs/integration/embeddings.md
• docs/integration/responses.md
• docs/integration/audio.md
• docs/integration/images.md
• docs/integration/rerank.md
• docs/integration/passthrough.md
• docs/integration/errors-and-retries.md

Acceptance Criteria

• The docs explain how to integrate with the real client-facing API surface
• OpenAI-compatible and Anthropic-style usage are both covered
• Provider-specific limitations are documented accurately
• Error semantics and retry guidance are customer-facing and actionable

[coderabbitai]

🔗 Related PRs

api7/ai-gateway-stash#72 - feat: anthropic provider [merged]
api7/api7ee-3-control-plane#2369 - docs: add hierarchical AGENTS.md knowledge base for AI coding agents [merged]
api7/docs#1431 - docs: add Anthropic-OpenAI protocol conversion guide for ai-proxy plugin [merged]
#14 - feat(provider): add new provider gateway types [merged]
api7/api7ee-3-control-plane#2609 - docs(AGENTS): require OpenAPI sync and prefer integration tests [merged]

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[moonming]

Closed by the merge of #249, which shipped openai-compatible-api.md (request-flow mermaid + endpoint surface + auth) and a code-verified errors-and-retries.md (error.type strings cross-checked against ProxyError). The other ten integration pages exist as placeholders and are tracked as a separate doc-refinement issue: #274.