[Push API] Fix flaky PushEventNotificationWithoutEventWaitUntil

chrome/browser/push_messaging/push_messaging_service_impl.cc

Index: chrome/browser/push_messaging/push_messaging_service_impl.cc
diff --git a/chrome/browser/push_messaging/push_messaging_service_impl.cc b/chrome/browser/push_messaging/push_messaging_service_impl.cc
index d90b5bd45f0c4340dfee977c126d8d401928e56a..9144cd5376fb74d6b36dcc1dd70379aff5c01690 100644
--- a/chrome/browser/push_messaging/push_messaging_service_impl.cc
+++ b/chrome/browser/push_messaging/push_messaging_service_impl.cc
@@ -395,15 +395,16 @@ void PushMessagingServiceImpl::DidHandleMessage(
   // an iterator rather than by value, as the latter removes all entries.
   in_flight_message_deliveries_.erase(in_flight_iterator);
 
-  message_handled_closure.Run();
-
-  if (push_messaging_service_observer_)
-    push_messaging_service_observer_->OnMessageHandled();
-
 #if BUILDFLAG(ENABLE_BACKGROUND)
+  // Reset before running callbacks below, so tests can verify keep-alive reset.
   if (in_flight_message_deliveries_.empty())
     in_flight_keep_alive_.reset();
 #endif
+
+  message_handled_closure.Run();
+
+  if (push_messaging_service_observer_)
+    push_messaging_service_observer_->OnMessageHandled();
 }
 
 void PushMessagingServiceImpl::SetMessageCallbackForTesting(