Reland of Change the update_client task runner behavior to continue on shutdown.

components/update_client/task_update.h

Index: components/update_client/task_update.h
diff --git a/components/update_client/task_update.h b/components/update_client/task_update.h
index 3d0963fb55d7eccadedb13e9e28723452a445ca4..aa3ce9d3b0f5645224e77df59b52cc5d96c87242 100644
--- a/components/update_client/task_update.h
+++ b/components/update_client/task_update.h
@@ -40,11 +40,14 @@
 
   void Run() override;
 
+  void Cancel() override;
+
   std::vector<std::string> GetIds() const override;
 
  private:
-  // Called when the Run function associated with this task has completed.
-  void RunComplete(int error);
+  // Called when the task has completed either because the task has run or
+  // it has been canceled.
+  void TaskComplete(int error);
 
   base::ThreadChecker thread_checker_;