WPF - Window DataContext or ElementName

WPF - Window DataContext or ElementName

I am currently facing a little problem with specifing or not specifing the
datacontext of a window, and why there is a difference between various
methods. Hope you can help me out.
Lets start with some code to show my problem. This is the code behind for
my TestWindow.xaml.cs, nothing really special about it just a simple
string property
public partial class TestWindow : Window
{
private string _helloWorld = "Hello World!";
public string HelloWorld
{
get { return _helloWorld; }
set { _helloWorld = value; }
}
public TestWindow()
{
InitializeComponent();
}
}
This code will be the same for all 3 following XAML layouts, so no changes
behind the code only in XAML.
1.) Databinding with given ElementName
<Window x:Class="Ktsw.Conx.ConxClient.TestWindow"
xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
Title="TestWindow" Height="300" Width="300"
Name="TestWin">
<Grid>
<TextBlock Text="{Binding HelloWorld,
ElementName=TestWin}"></TextBlock>
</Grid>
</Window>
2.) Databinding with specifing DataContext on Window
<Window x:Class="Ktsw.Conx.ConxClient.TestWindow"
xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
Title="TestWindow" Height="300" Width="300"
DataContext="{Binding RelativeSource={RelativeSource Self}}">
<Grid>
<TextBlock Text="{Binding HelloWorld}"></TextBlock>
</Grid>
</Window>
3.) Neither ElementName nor specifing DataContext
<Window x:Class="Ktsw.Conx.ConxClient.TestWindow"
xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
Title="TestWindow" Height="300" Width="300">
<Grid>
<TextBlock Text="{Binding HelloWorld}"></TextBlock>
</Grid>
</Window>
The first two methods work just fine, but why fails the 3rd?
In the first method I am not specifing the DataContext and it works
automatically, in the second method I am not specifing the ElementName and
it works, but without declaring one of those it fails. Why would it fail
getting both automatically, but work fine with getting each individually?