[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/core/src/tools/router_tests.rs

@@ -25,18 +25,12 @@ async fn js_repl_tools_only_blocks_direct_tool_calls() -> anyhow::Result<()> {
         .await
         .list_all_tools()
         .await;
-    let app_tools = Some(mcp_tools.clone());
+    let deferred_mcp_tools = Some(mcp_tools.clone());
     let router = ToolRouter::from_config(
         &turn.tools_config,
         ToolRouterParams {
-            mcp_tools: Some(
-                mcp_tools
-                    .into_iter()
-                    .map(|(name, tool)| (name, tool.tool))
-                    .collect(),
-            ),
-            tool_namespaces: None,
-            app_tools,
+            deferred_mcp_tools,
+            mcp_tools: Some(mcp_tools),
             discoverable_tools: None,
             dynamic_tools: turn.dynamic_tools.as_slice(),
         },
@@ -84,18 +78,12 @@ async fn js_repl_tools_only_allows_js_repl_source_calls() -> anyhow::Result<()>
         .await
         .list_all_tools()
         .await;
-    let app_tools = Some(mcp_tools.clone());
+    let deferred_mcp_tools = Some(mcp_tools.clone());
     let router = ToolRouter::from_config(
         &turn.tools_config,
         ToolRouterParams {
-            mcp_tools: Some(
-                mcp_tools
-                    .into_iter()
-                    .map(|(name, tool)| (name, tool.tool))
-                    .collect(),
-            ),
-            tool_namespaces: None,
-            app_tools,
+            deferred_mcp_tools,
+            mcp_tools: Some(mcp_tools),
             discoverable_tools: None,
             dynamic_tools: turn.dynamic_tools.as_slice(),
         },