[codex] Add response.processed websocket request (#21284)

## Summary

- Add a `response.processed` websocket request payload and sender for
Responses API websockets.
- Send `response.processed` from `try_run_sampling_request` after a
response completes, local turn processing succeeds, and the
session-owned feature flag is enabled.
- Add websocket coverage for both enabled and disabled feature-flag
behavior.

## Validation

- `just fmt`
- `cargo test -p codex-core response_processed`
- `cargo test -p codex-api responses_websocket`
- `cargo test -p codex-features
responses_websocket_response_processed_is_under_development`
- `git diff --check`
- `just fix -p codex-api -p codex-core -p codex-features`
- `git diff --check origin/main...HEAD`

codex-rs/core/src/session/turn.rs

@@ -1884,6 +1884,7 @@ async fn try_run_sampling_request(
     let mut assistant_message_stream_parsers = AssistantMessageStreamParsers::new(plan_mode);
     let mut plan_mode_state = plan_mode.then(|| PlanModeStreamState::new(&turn_context.sub_id));
     let receiving_span = trace_span!("receiving_stream");
+    let mut completed_response_id: Option<String> = None;
     let outcome: CodexResult<SamplingRequestResult> = loop {
         let handle_responses = trace_span!(
             parent: &receiving_span,
@@ -2111,7 +2112,7 @@ async fn try_run_sampling_request(
                 sess.services.models_manager.refresh_if_new_etag(etag).await;
             }
             ResponseEvent::Completed {
-                response_id: _,
+                response_id,
                 token_usage,
                 end_turn,
             } => {
@@ -2128,6 +2129,7 @@ async fn try_run_sampling_request(
                 if let Some(false) = end_turn {
                     needs_follow_up = true;
                 }
+                completed_response_id = Some(response_id);
                 break Ok(SamplingRequestResult {
                     needs_follow_up,
                     last_agent_message,
@@ -2239,6 +2241,15 @@ async fn try_run_sampling_request(
     )
     .await;
 
+    if sess
+        .features
+        .enabled(Feature::ResponsesWebsocketResponseProcessed)
+        && outcome.is_ok()
+        && let Some(response_id) = completed_response_id.as_deref()
+    {
+        client_session.send_response_processed(response_id).await;
+    }
+
     drain_in_flight(&mut in_flight, sess.clone(), turn_context.clone()).await?;
 
     if cancellation_token.is_cancelled() {