Add indexed web search mode (#28489)

## Summary

- Add `web_search = "indexed"` alongside `disabled`, `cached`, and
`live`.
- Use that same resolved mode for both hosted and standalone web search.
- For hosted search, send `index_gated_web_access: true` with external
web access enabled only when `indexed` is selected.
- For standalone search, preserve the existing boolean wire values for
existing modes (`cached` maps to `false` and `live` to `true`) and send
`"indexed"` only for `indexed`; `disabled` keeps the tool unavailable.
- Carry the mode through managed configuration requirements and
generated schemas.

## Why

Indexed search provides a middle ground between cached-only search and
unrestricted live page fetching. Search queries can remain live while
direct page fetches are limited to URLs admitted by the server.

The existing `web_search` setting remains the single source of truth, so
hosted and standalone executors cannot drift into different access
modes. Without an explicit `indexed` selection, the existing
model-visible tool and request shapes are unchanged.

```toml
web_search = "indexed"

[features]
standalone_web_search = true
```

## Validation

- `just fmt`
- `just test -p codex-api` (`126 passed`)
- `just test -p codex-web-search-extension` (`7 passed`)
- `just test -p codex-core
code_mode_can_call_indexed_standalone_web_search` (`1 passed`)
- Focused configuration, hosted request, standalone request, and
managed-requirement coverage is included in the PR; remaining suites run
in CI.

The full workspace test suite was not run locally.

codex-rs/app-server-protocol/schema/json/codex_app_server_protocol.v2.schemas.json

@@ -18134,6 +18134,7 @@
       "enum": [
         "disabled",
         "cached",
+        "indexed",
         "live"
       ],
       "type": "string"