Fix Codex Apps auth elicitation hang (#29615)

## Summary
- Require the reserved Codex Apps MCP server name to be present in the
connection manager before treating it as host-owned.
- Update auth elicitation tests to model an installed host-owned Codex
Apps server without sending startup events to the test session.

## Why
PR #29518 replaced the old host-owned flag with a name-only check. That
made non-host-owned tests with the reserved codex_apps name enter auth
elicitation and wait forever for a response.

codex-rs/codex-mcp/src/connection_manager_tests.rs

@@ -1177,6 +1177,41 @@ fn server_metadata_preserves_tool_approval_policy() {
     );
 }
 
+#[test]
+fn host_owned_codex_apps_requires_server_metadata() {
+    let approval_policy = Constrained::allow_any(AskForApproval::OnFailure);
+    let permission_profile = Constrained::allow_any(PermissionProfile::default());
+    let manager = McpConnectionManager::new_uninitialized(
+        &approval_policy,
+        &permission_profile,
+        /*prefix_mcp_tool_names*/ true,
+    );
+
+    assert!(!manager.is_host_owned_codex_apps_server(CODEX_APPS_MCP_SERVER_NAME));
+}
+
+#[test]
+fn host_owned_codex_apps_matches_reserved_name_with_server_metadata() {
+    let approval_policy = Constrained::allow_any(AskForApproval::OnFailure);
+    let permission_profile = Constrained::allow_any(PermissionProfile::default());
+    let mut manager = McpConnectionManager::new_uninitialized(
+        &approval_policy,
+        &permission_profile,
+        /*prefix_mcp_tool_names*/ true,
+    );
+    let server = EffectiveMcpServer::configured(crate::codex_apps_mcp_server_config(
+        "https://chatgpt.com",
+        /*apps_mcp_product_sku*/ None,
+    ));
+    manager.server_metadata.insert(
+        CODEX_APPS_MCP_SERVER_NAME.to_string(),
+        McpServerMetadata::from(&server),
+    );
+
+    assert!(manager.is_host_owned_codex_apps_server(CODEX_APPS_MCP_SERVER_NAME));
+    assert!(!manager.is_host_owned_codex_apps_server("docs"));
+}
+
 #[tokio::test]
 async fn no_local_runtime_fails_local_stdio_but_keeps_local_http_server() {
     let approval_policy = Constrained::allow_any(AskForApproval::OnFailure);