feat: do not close unified exec processes across turns (#10799)

With this PR we do not close the unified exec processes (i.e. background
terminals) at the end of a turn unless:
* The user interrupt the turn
* The user decide to clean the processes through `app-server` or
`/clean`

I made sure that `codex exec` correctly kill all the processes

codex-rs/app-server-protocol/src/protocol/common.rs

@@ -214,6 +214,11 @@ client_request_definitions! {
         params: v2::ThreadCompactStartParams,
         response: v2::ThreadCompactStartResponse,
     },
+    #[experimental("thread/backgroundTerminals/clean")]
+    ThreadBackgroundTerminalsClean => "thread/backgroundTerminals/clean" {
+        params: v2::ThreadBackgroundTerminalsCleanParams,
+        response: v2::ThreadBackgroundTerminalsCleanResponse,
+    },
     ThreadRollback => "thread/rollback" {
         params: v2::ThreadRollbackParams,
         response: v2::ThreadRollbackResponse,
@@ -1120,6 +1125,27 @@ mod tests {
         Ok(())
     }
 
+    #[test]
+    fn serialize_thread_background_terminals_clean() -> Result<()> {
+        let request = ClientRequest::ThreadBackgroundTerminalsClean {
+            request_id: RequestId::Integer(8),
+            params: v2::ThreadBackgroundTerminalsCleanParams {
+                thread_id: "thr_123".to_string(),
+            },
+        };
+        assert_eq!(
+            json!({
+                "method": "thread/backgroundTerminals/clean",
+                "id": 8,
+                "params": {
+                    "threadId": "thr_123"
+                }
+            }),
+            serde_json::to_value(&request)?,
+        );
+        Ok(())
+    }
+
     #[test]
     fn mock_experimental_method_is_marked_experimental() {
         let request = ClientRequest::MockExperimentalMethod {