fix(tool): support Codex web search GPT-5.6 Responses-Lite models

[roboomp]

Repro

With PI_CODEX_WEB_SEARCH_MODEL=gpt-5.6-luna, a mocked searchCodex call against packages/coding-agent/src/web/search/providers/codex.ts captured a /backend-api/codex/responses request that still had top-level tools, tool_choice, and instructions, began input with the user message, and omitted x-openai-internal-codex-responses-lite: true. The default model preference chain also started at gpt-5.5, so default Codex web search never tried Luna, Terra, or Sol first.

Cause

packages/coding-agent/src/web/search/providers/codex.ts hard-coded DEFAULT_MODEL_PREFERENCES to GPT-5.5-and-older models and built one classic Responses request body in callCodexSearch for every model. The provider never consulted bundled Codex model metadata, so useResponsesLite: true on the GPT-5.6 catalog entries could not affect request encoding.

Fix

• Added GPT-5.6 Luna, Terra, and Sol to the Codex web-search default preference chain before GPT-5.5.
• Resolved bundled Codex model metadata for default and explicit PI_CODEX_WEB_SEARCH_MODEL requests, including validated <model>:<effort> suffixes.
• Switched only useResponsesLite models to developer additional_tools, developer instructions, Responses-Lite headers, Codex compatibility metadata, and reasoning.context=all_turns; classic models keep the existing top-level tools, tool_choice, and instructions shape.
• Added focused regression coverage for GPT-5.6 default ordering, default fallback, explicit override behavior, Responses-Lite encoding, effort suffixes, and classic GPT-5.4 shape preservation.
• Added a packages/coding-agent/CHANGELOG.md entry for the fix.

Verification

bun test packages/coding-agent/test/tools/web-search-codex.test.ts passed: 14 pass, 0 fail, 54 expect() calls. bun run check:ts passed across the TypeScript workspaces. bun check fails on main for an unrelated Rust formatting diff in crates/pi-ast/src/ops.rs; git diff --name-only origin/main..HEAD -- crates/pi-ast/src/ops.rs is empty, so the pre-publish gate was skipped. Fixes #5286