public partial class MainWindow : Window
{
    private CancellationTokenSource tokenSource = new CancellationTokenSource();

    public MainWindow()
    {
        InitializeComponent();
        PrintButtonText("None");
    }

    private void PrintButtonText(string buttonText)
    {
        Console.WriteLine("Update!");
        Task.Factory.StartNew(() =>
        {
            while (!tokenSource.Token.IsCancellationRequested)
            {
                Console.WriteLine("Button Pressed Text: " + buttonText);
            }
        }, tokenSource.Token);
    }

    private void Button1_Click(object sender, RoutedEventArgs e)
    {
        tokenSource.Cancel();
        PrintButtonText("Button1");
    }

    private void Button2_Click(object sender, RoutedEventArgs e)
    {
        tokenSource.Cancel();
        PrintButtonText("Button2");
    }
}

tokenSource.Cancel();
PrintButtonText("Button1");