feat: add header for task kind (#5142)

Add a header in the responses API request for the task kind (compact,
review, ...) for observability purpose
The header name is `codex-task-type`

codex-rs/core/src/tasks/regular.rs

@@ -27,6 +27,6 @@ impl SessionTask for RegularTask {
         input: Vec<InputItem>,
     ) -> Option<String> {
         let sess = session.clone_session();
-        run_task(sess, ctx, sub_id, input).await
+        run_task(sess, ctx, sub_id, input, TaskKind::Regular).await
     }
 }

codex-rs/core/src/tasks/review.rs

@@ -28,7 +28,7 @@ impl SessionTask for ReviewTask {
         input: Vec<InputItem>,
     ) -> Option<String> {
         let sess = session.clone_session();
-        run_task(sess, ctx, sub_id, input).await
+        run_task(sess, ctx, sub_id, input, TaskKind::Review).await
     }
 
     async fn abort(&self, session: Arc<SessionTaskContext>, sub_id: &str) {