WPF Rounded border button responsive to click color change

I have a button on my WPF application (MVVM Pattern) which is responsive to a click event. Basically if you click on this button its background becomes LightGreen (the default color is LightGray). I already achieved the desired behavior using the following code:

<Button Grid.Column="2" Grid.Row="6" Grid.ColumnSpan="3" Grid.RowSpan="2" Content="{Binding FirstSchedule.Message}" Command="{Binding FirstScheduleButtonClick}">
    <Button.Style>
        <Style TargetType="Button">
            <Setter Property="Background" Value="LightGray"></Setter>
            <Style.Triggers>
                <DataTrigger Binding="{Binding FirstScheduleButtonSelected}" Value="True">
                    <Setter Property="Background" Value="LightGreen"></Setter>
                </DataTrigger>
            </Style.Triggers>
        </Style>
    </Button.Style>
</Button>

In which FirstScheduleButtonSelected is a ViewModel property defined by:

private bool _firstScheduleButtonSelected;
public bool FirstScheduleButtonSelected
{
    get { return _firstScheduleButtonSelected; }
    set { _firstScheduleButtonSelected = value;  NotifyPropertyChanged("FirstScheduleButtonSelected"); NotifyPropertyChanged("Background"); }
}

Now I need to make this button's borders rounded. I already tried this solution How to create/make rounded corner buttons in WPF?, and I ended up with:

<Button Grid.Column="2" Grid.Row="6" Grid.ColumnSpan="3" Grid.RowSpan="2" Content="{Binding FirstSchedule.Message}" Command="{Binding FirstScheduleButtonClick}">
    <Button.Style>
        <Style TargetType="Button">
            <Setter Property="Template">
                <Setter.Value>
                    <ControlTemplate TargetType="Button">
                        <Border CornerRadius="15" Background="LightGray" BorderThickness="1" Padding="2">
                            <ContentPresenter HorizontalAlignment="Center" VerticalAlignment="Center" />
                        </Border>
                    </ControlTemplate>
                </Setter.Value>
            </Setter>
            <Setter Property="Background" Value="LightGray"></Setter>
            <Style.Triggers>
                <DataTrigger Binding="{Binding FirstScheduleButtonSelected}" Value="True">
                    <Setter Property="Background" Value="LightGreen"></Setter>
                </DataTrigger>
            </Style.Triggers>
        </Style>
    </Button.Style>
</Button>

Now my borders are, in fact, rounded but when I click the button it does not becomes green.

Q: How can I modify this button in order to make its borders rounded and keep my already functioning behavior of changing its color on click?