backend: add provider-status example response to docs

Goal

Document three representative /v1/providers/status response excerpts so a contributor can understand the local-first provider boundary without configuring paid APIs or local speech engines.

Scope

• Edit only docs/backend/API_RUNBOOK.md.
• Add a short provider-status subsection near the existing quick check or Provider adapters section.
• Include three trimmed JSON examples:
  1. Default mock mode: mode: "mock_open_core" with providers.llm, providers.tts, and providers.stt showing active: "mock", externalConfigured: false, and replaceable: true.
  2. Fallback mode: operations.rateLimit showing backend: "memory", requestedBackend: "redis", and an illustrative fallbackReason when Redis was requested but unavailable.
  3. Local speech engines: providers.tts showing active: "voicevox_compat" with its engine and fallback, plus providers.stt showing active: "whisper_cpp" and its boolean readiness fields.
• Keep examples intentionally trimmed. /v1/providers/status also reports broader auth/content/experiment capabilities; this issue does not need to reproduce the entire payload.
• Use apps/api/app/main.py, apps/api/app/providers.py, and apps/api/app/rate_limit.py only as sources of truth. Do not edit production code for this issue.
• Use placeholders for machine-specific paths, exception classes, and base URLs. Never include a real API key, token, private host, or local user path.

Acceptance Criteria

• All three labeled examples are present and valid JSON after replacing explanatory placeholders with JSON strings.
• Field names and nesting match the current implementation.
• The surrounding text explains that externalApiKeysRequired is false and that fallback status is diagnostic rather than proof that a live external engine succeeded.
• The change stays within docs/backend/API_RUNBOOK.md.
• The PR body includes Closes #19.

Suggested Verification

This is a browser-editable docs task; no backend or speech-engine installation is required.

python3 scripts/check_public_tree.py
git diff --check

Direct edit link: https://github.com/duct-tape2/ai-language-partner/edit/main/docs/backend/API_RUNBOOK.md