[codex] Support search schema on all search types

[LiamHz]

What changed

This updates the JavaScript SDK search surface so systemPrompt and outputSchema are supported across all search types, adds deep-lite as a deep search variant, and documents that output is returned when outputSchema is provided.

Why

The search API now supports these parameters across every search type instead of only deep and deep-reasoning, and it adds deep-lite alongside the existing deep variants.

Impact

JavaScript SDK consumers can now pass systemPrompt and outputSchema on non-deep searches, use deep-lite, and rely on the documented output field when structured output is requested.

Validation

• npx vitest run test/unit/search.test.ts
• npm run build
• npm run typecheck still has unrelated pre-existing repo errors outside this changed surface

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[devin-ai-integration]

✅ Devin Review: No Issues Found

Devin Review analyzed this PR and found no potential bugs to report.

View in Devin Review to see 4 additional findings.