Fix stale background terminal poll events (#23231)

## Why

Issue #23214 reports `/ps` showing no background terminals while the
status line still says it is waiting for a background terminal. The race
is in core: `write_stdin` can poll a process that exits before the
response returns. The process manager correctly returns `process_id:
None`, but the handler still emitted a `TerminalInteraction` event using
the requested session id, causing clients to believe a dead process was
still being polled.

Fixes #23214.

## What changed

- Suppress `TerminalInteraction` events for empty `write_stdin` polls
once `response.process_id` is `None`.
- Continue emitting interactions for non-empty stdin, even if that input
causes the process to exit before the response returns.
- Extend the unified exec integration test to assert completed empty
polls do not emit terminal interactions.

## Verification

- `cargo test -p codex-core --test all
unified_exec_emits_one_begin_and_one_end_event`
- `cargo test -p codex-core --test all
unified_exec_emits_terminal_interaction_for_write_stdin`

`cargo test -p codex-core` currently aborts in unrelated
`agent::control::tests::resume_agent_from_rollout_uses_edge_data_when_descendant_metadata_source_is_stale`
with a reproducible stack overflow.

codex-rs/core/src/tools/handlers/unified_exec/write_stdin.rs

@@ -78,14 +78,21 @@ impl ToolExecutor<ToolInvocation> for WriteStdinHandler {
                 FunctionCallError::RespondToModel(format!("write_stdin failed: {err}"))
             })?;
 
-        let interaction = TerminalInteractionEvent {
-            call_id: response.event_call_id.clone(),
-            process_id: args.session_id.to_string(),
-            stdin: args.chars.clone(),
-        };
-        session
-            .send_event(turn.as_ref(), EventMsg::TerminalInteraction(interaction))
-            .await;
+        // Empty stdin is a background poll, so emit it only while there is
+        // still a live process for the UI to wait on. Non-empty stdin is a real
+        // terminal interaction and should remain visible even if it completes
+        // the process before the response returns.
+        if !args.chars.is_empty() || response.process_id.is_some() {
+            let process_id = response.process_id.unwrap_or(args.session_id);
+            let interaction = TerminalInteractionEvent {
+                call_id: response.event_call_id.clone(),
+                process_id: process_id.to_string(),
+                stdin: args.chars.clone(),
+            };
+            session
+                .send_event(turn.as_ref(), EventMsg::TerminalInteraction(interaction))
+                .await;
+        }
 
         Ok(boxed_tool_output(response))
     }

codex-rs/core/src/unified_exec/process_manager.rs

@@ -691,8 +691,8 @@ impl UnifiedExecProcessManager {
 
         // After polling, refresh_process_state tells us whether the PTY is
         // still alive or has exited and been removed from the store; we thread
-        // that through so the handler can tag TerminalInteraction with an
-        // appropriate process_id and exit_code.
+        // that through so the handler can tag or suppress TerminalInteraction
+        // with an appropriate process_id and exit_code.
         let status = if let Some(status) = status_after_write {
             status
         } else {