Expose MCP server info as part of server status (#24698)

# Summary

Expose MCP server info via App Server (when available) so apps can
render a richer MCP experience

codex-rs/app-server/tests/suite/v2/mcp_server_status.rs

@@ -20,6 +20,7 @@ use codex_core::config::set_project_trust_level;
 use codex_protocol::config_types::TrustLevel;
 use pretty_assertions::assert_eq;
 use rmcp::handler::server::ServerHandler;
+use rmcp::model::Implementation;
 use rmcp::model::JsonObject;
 use rmcp::model::ListResourceTemplatesResult;
 use rmcp::model::ListResourcesResult;
@@ -99,6 +100,13 @@ url = "{mcp_server_url}/mcp"
             .map(|tool| tool.name.as_str()),
         Some("look-up.raw")
     );
+    assert_eq!(
+        status
+            .server_info
+            .as_ref()
+            .and_then(|info| info.title.as_deref()),
+        Some("Lookup Server")
+    );
 
     mcp_server_handle.abort();
     let _ = mcp_server_handle.await;
@@ -205,7 +213,9 @@ struct McpStatusServer {
 
 impl ServerHandler for McpStatusServer {
     fn get_info(&self) -> ServerInfo {
-        ServerInfo::new(ServerCapabilities::builder().enable_tools().build())
+        ServerInfo::new(ServerCapabilities::builder().enable_tools().build()).with_server_info(
+            Implementation::new("lookup-server", "1.0.0").with_title("Lookup Server"),
+        )
     }
 
     async fn list_tools(