[mcp] Expand tool search to custom MCPs. (#16944)

- [x] Expand tool search to custom MCPs.
- [x] Rename several variables/fields to be more generic.

Updated tool & server name lifecycles:

**Raw Identity**

ToolInfo.server_name is raw MCP server name.
ToolInfo.tool.name is raw MCP tool name.
MCP calls route back to raw via parse_tool_name() returning
(tool.server_name, tool.tool.name).
mcpServerStatus/list now groups by raw server and keys tools by
Tool.name: mod.rs:599
App-server just forwards that grouped raw snapshot:
codex_message_processor.rs:5245

**Callable Names**

On list-tools, we create provisional callable_namespace / callable_name:
mcp_connection_manager.rs:1556
For non-app MCP, provisional callable name starts as raw tool name.
For codex-apps, provisional callable name is sanitized and strips
connector name/id prefix; namespace includes connector name.
Then qualify_tools() sanitizes callable namespace + name to ASCII alnum
/ _ only: mcp_tool_names.rs:128
Note: this is stricter than Responses API. Hyphen is currently replaced
with _ for code-mode compatibility.

**Collision Handling**

We do initially collapse example-server and example_server to the same
base.
Then qualify_tools() detects distinct raw namespace identities behind
the same sanitized namespace and appends a hash to the callable
namespace: mcp_tool_names.rs:137
Same idea for tool-name collisions: hash suffix goes on callable tool
name.
Final list_all_tools() map key is callable_namespace + callable_name:
mcp_connection_manager.rs:769

**Direct Model Tools**

Direct MCP tool declarations use the full qualified sanitized key as the
Responses function name.
The raw rmcp Tool is converted but renamed for model exposure.

**Tool Search / Deferred**

Tool search result namespace = final ToolInfo.callable_namespace:
tool_search.rs:85
Tool search result nested name = final ToolInfo.callable_name:
tool_search.rs:86
Deferred tool handler is registered as "{namespace}:{name}":
tool_registry_plan.rs:248
When a function call comes back, core recombines namespace + name, looks
up the full qualified key, and gets the raw server/tool for MCP
execution: codex.rs:4353

**Separate Legacy Snapshot**

collect_mcp_snapshot_from_manager_with_detail() still returns a map
keyed by qualified callable name.
mcpServerStatus/list no longer uses that; it uses
McpServerStatusSnapshot, which is raw-inventory shaped.

codex-rs/tools/src/lib.rs

@@ -101,12 +101,12 @@ pub use tool_discovery::DiscoverableToolType;
 pub use tool_discovery::TOOL_SEARCH_DEFAULT_LIMIT;
 pub use tool_discovery::TOOL_SEARCH_TOOL_NAME;
 pub use tool_discovery::TOOL_SUGGEST_TOOL_NAME;
-pub use tool_discovery::ToolSearchAppInfo;
-pub use tool_discovery::ToolSearchAppSource;
 pub use tool_discovery::ToolSearchResultSource;
+pub use tool_discovery::ToolSearchSource;
+pub use tool_discovery::ToolSearchSourceInfo;
 pub use tool_discovery::ToolSuggestEntry;
-pub use tool_discovery::collect_tool_search_app_infos;
 pub use tool_discovery::collect_tool_search_output_tools;
+pub use tool_discovery::collect_tool_search_source_infos;
 pub use tool_discovery::collect_tool_suggest_entries;
 pub use tool_discovery::create_tool_search_tool;
 pub use tool_discovery::create_tool_suggest_tool;
@@ -116,7 +116,7 @@ pub use tool_registry_plan_types::ToolHandlerKind;
 pub use tool_registry_plan_types::ToolHandlerSpec;
 pub use tool_registry_plan_types::ToolNamespace;
 pub use tool_registry_plan_types::ToolRegistryPlan;
-pub use tool_registry_plan_types::ToolRegistryPlanAppTool;
+pub use tool_registry_plan_types::ToolRegistryPlanDeferredTool;
 pub use tool_registry_plan_types::ToolRegistryPlanParams;
 pub use tool_spec::ConfiguredToolSpec;
 pub use tool_spec::ResponsesApiWebSearchFilters;