A StaticResource will be resolved and assigned to the property during the loading of the XAML which occurs before the application is actually run. It will only be assigned once and any changes to resource dictionary ignored.
A DynamicResource assigns an Expression object to the property during loading but does not actually lookup the resource until runtime when the Expression object is asked for the value. This defers looking up the resource until it is needed at runtime. A good example would be a forward reference to a resource defined later on in the XAML. Another example is a resource that will not even exist until runtime. It will update the target if the source resource dictionary is changed.
There really is only one name in XAML, the x:Name. A framework, such as WPF, can optionally map one of its properties to XAML's x:Name by using the RuntimeNamePropertyAttribute on the class that designates one of the classes properties as mapping to the x:Name attribute of XAML.
The reason this was done was to allow for frameworks that already have a concept of "Name" at runtime, such as WPF. In WPF, for example, FrameworkElement introduces a Name property.
In general, a class does not need to store the name for x:Name to be useable. All x:Name means to XAML is generate a field to store the value in the code behind class. What the runtime does with that mapping is framework dependent.
So, why are there two ways to do the same thing? The simple answer is because there are two concepts mapped onto one property. WPF wants the name of an element preserved at runtime (which is usable through Bind, among other things) and XAML needs to know what elements you want to be accessible by fields in the code behind class. WPF ties these two together by marking the Name property as an alias of x:Name.
In the future, XAML will have more uses for x:Name, such as allowing you to set properties by referring to other objects by name, but in 3.5 and prior, it is only used to create fields.
Whether you should use one or the other is really a style question, not a technical one. I will leave that to others for a recommendation.
See also AutomationProperties.Name VS x:Name, AutomationProperties.Name is used by accessibility tools and some testing tools.
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