[fix] app server flaky send_messages test (#8874)

Fix flakiness of CI test:
https://github.com/openai/codex/actions/runs/20350530276/job/58473691434?pr=8282

This PR does two things:
1. move the flakiness test to use responses API instead of chat
completion API
2. make mcp_process agnostic to the order of
responses/notifications/requests that come in, by buffering messages not
read

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

@@ -554,6 +554,7 @@ async fn turn_start_updates_sandbox_and_cwd_between_turns_v2() -> Result<()> {
         mcp.read_stream_until_notification_message("codex/event/task_complete"),
     )
     .await??;
+    mcp.clear_message_buffer();
 
     // second turn with workspace-write and second_cwd, ensure exec begins in second_cwd
     let second_turn = mcp