fix(process): label background completion causes (#46659)

Track why a background process finished and include that source in notify-on-complete messages so SIGTERM from process.kill, kill_all, backend loss, and ordinary exits are distinguishable.

tests/tools/test_process_registry.py

@@ -1012,12 +1012,41 @@ def test_format_completion_event():
         "output": "done",
     }
     result = format_process_notification(evt)
-    assert "[IMPORTANT: Background process proc_abc completed" in result
+    assert "[IMPORTANT: Background process proc_abc completed normally" in result
     assert "exit code 0" in result
     assert "Command: sleep 5" in result
     assert "Output:\ndone]" in result
 
 
+def test_format_killed_completion_event_names_source_and_signal():
+    evt = {
+        "type": "completion",
+        "session_id": "proc_killed",
+        "command": "sleep 5",
+        "exit_code": -15,
+        "completion_reason": "killed",
+        "termination_source": "process.kill",
+        "output": "",
+    }
+    result = format_process_notification(evt)
+    assert "proc_killed terminated by process.kill" in result
+    assert "exit code -15, SIGTERM" in result
+
+
+def test_format_external_sigterm_does_not_claim_agent_kill():
+    evt = {
+        "type": "completion",
+        "session_id": "proc_external",
+        "command": "sleep 5",
+        "exit_code": 143,
+        "output": "",
+    }
+    result = format_process_notification(evt)
+    assert "proc_external exited" in result
+    assert "terminated by" not in result
+    assert "exit code 143, SIGTERM" in result
+
+
 def test_format_watch_match_event():
     evt = {
         "type": "watch_match",
@@ -1087,7 +1116,7 @@ def test_drain_notifications_returns_pending_events():
         results = process_registry.drain_notifications()
         assert len(results) == 2
         assert results[0][0]["session_id"] == "proc_drain1"
-        assert "proc_drain1 completed" in results[0][1]
+        assert "proc_drain1 completed normally" in results[0][1]
         assert results[1][0]["session_id"] == "proc_drain2"
         assert "watch pattern" in results[1][1]
     finally: