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/src/lib.rs

@@ -48,7 +48,7 @@ use codex_core::default_client::set_default_originator;
 use codex_core::find_conversation_path_by_id_str;
 
 pub async fn run_main(cli: Cli, codex_linux_sandbox_exe: Option<PathBuf>) -> anyhow::Result<()> {
-    if let Err(err) = set_default_originator("codex_exec") {
+    if let Err(err) = set_default_originator("codex_exec".to_string()) {
         tracing::warn!(?err, "Failed to set codex exec originator override {err:?}");
     }