Encapsulate tool search entries in handlers (#22261)

## Why

This builds on the handler-owned spec refactor by moving deferred
tool-search metadata to the same handlers that already own tool specs.
The registry builder no longer needs a separate prebuilt
`tool_search_entries` path; it can collect searchable entries from
deferred handlers directly.

## What changed

- Added `search_info()` to tool handlers and implemented it for MCP and
dynamic handlers.
- Reused handler `spec()` output when constructing tool-search entries,
adapting it into the deferred `LoadableToolSpec` shape expected by
`tool_search`.
- Simplified `build_tool_registry_builder(...)` so `tool_search`
registration is based on deferred handlers with search info.
- Removed the old standalone search-entry builders and now-unused
`codex-tools` discovery helper exports.

## Verification

- `cargo test -p codex-core tools::handlers::tool_search::tests:: --
--nocapture`
- `cargo test -p codex-core tools::spec_plan::tests::search_tool --
--nocapture`
- `cargo test -p codex-core tools::spec::tests:: -- --nocapture`
- `cargo test -p codex-core tools::spec_plan::tests:: -- --nocapture`
- `cargo test -p codex-tools`
- `just fix -p codex-core`
- `just fix -p codex-tools`

codex-rs/tools/src/lib.rs

@@ -47,7 +47,6 @@ pub use responses_api::ResponsesApiNamespaceTool;
 pub use responses_api::ResponsesApiTool;
 pub use responses_api::coalesce_loadable_tool_specs;
 pub use responses_api::default_namespace_description;
-pub use responses_api::dynamic_tool_to_loadable_tool_spec;
 pub use responses_api::dynamic_tool_to_responses_api_tool;
 pub use responses_api::mcp_tool_to_deferred_responses_api_tool;
 pub use responses_api::mcp_tool_to_responses_api_tool;
@@ -69,13 +68,9 @@ pub use tool_discovery::REQUEST_PLUGIN_INSTALL_TOOL_NAME;
 pub use tool_discovery::RequestPluginInstallEntry;
 pub use tool_discovery::TOOL_SEARCH_DEFAULT_LIMIT;
 pub use tool_discovery::TOOL_SEARCH_TOOL_NAME;
-pub use tool_discovery::ToolSearchResultSource;
-pub use tool_discovery::ToolSearchSource;
 pub use tool_discovery::ToolSearchSourceInfo;
 pub use tool_discovery::collect_request_plugin_install_entries;
-pub use tool_discovery::collect_tool_search_source_infos;
 pub use tool_discovery::filter_request_plugin_install_discoverable_tools_for_client;
-pub use tool_discovery::tool_search_result_source_to_loadable_tool_spec;
 pub use tool_spec::ResponsesApiWebSearchFilters;
 pub use tool_spec::ResponsesApiWebSearchUserLocation;
 pub use tool_spec::ToolSpec;