Set codex SDK TypeScript originator (#4894)

## Summary
- ensure the TypeScript SDK sets CODEX_INTERNAL_ORIGINATOR_OVERRIDE to
codex_sdk_ts when spawning the Codex CLI
- extend the responses proxy test helper to capture request headers for
assertions
- add coverage that verifies Codex threads launched from the TypeScript
SDK send the codex_sdk_ts originator header

## Testing
- Not Run (not requested)


------
https://chatgpt.com/codex/tasks/task_i_68e561b125248320a487f129093d16e7

codex-rs/exec/tests/suite/mod.rs

@@ -1,6 +1,7 @@
 // Aggregates all former standalone integration tests as modules.
 mod apply_patch;
 mod auth_env;
+mod originator;
 mod output_schema;
 mod resume;
 mod sandbox;

codex-rs/exec/tests/suite/originator.rs

@@ -0,0 +1,52 @@
+#![cfg(not(target_os = "windows"))]
+#![allow(clippy::expect_used, clippy::unwrap_used)]
+
+use core_test_support::responses;
+use core_test_support::test_codex_exec::test_codex_exec;
+use wiremock::matchers::header;
+
+/// Verify that when the server reports an error, `codex-exec` exits with a
+/// non-zero status code so automation can detect failures.
+#[tokio::test(flavor = "multi_thread", worker_threads = 2)]
+async fn send_codex_exec_originator() -> anyhow::Result<()> {
+    let test = test_codex_exec();
+
+    let server = responses::start_mock_server().await;
+    let body = responses::sse(vec![
+        responses::ev_response_created("response_1"),
+        responses::ev_assistant_message("response_1", "Hello, world!"),
+        responses::ev_completed("response_1"),
+    ]);
+    responses::mount_sse_once_match(&server, header("Originator", "codex_exec"), body).await;
+
+    test.cmd_with_server(&server)
+        .arg("--skip-git-repo-check")
+        .arg("tell me something")
+        .assert()
+        .code(0);
+
+    Ok(())
+}
+
+#[tokio::test(flavor = "multi_thread", worker_threads = 2)]
+async fn supports_originator_override() -> anyhow::Result<()> {
+    let test = test_codex_exec();
+
+    let server = responses::start_mock_server().await;
+    let body = responses::sse(vec![
+        responses::ev_response_created("response_1"),
+        responses::ev_assistant_message("response_1", "Hello, world!"),
+        responses::ev_completed("response_1"),
+    ]);
+    responses::mount_sse_once_match(&server, header("Originator", "codex_exec_override"), body)
+        .await;
+
+    test.cmd_with_server(&server)
+        .env("CODEX_INTERNAL_ORIGINATOR_OVERRIDE", "codex_exec_override")
+        .arg("--skip-git-repo-check")
+        .arg("tell me something")
+        .assert()
+        .code(0);
+
+    Ok(())
+}