Why Local Functions generate IL different from Anonymous Methods and Lambda Expressions?

Why the C# 7 Compiler turns Local Functions into methods within the same class where their parent function is. While for Anonymous Methods (and Lambda Expressions) the compiler generates a nested class for each parent function, that will contain all of its Anonymous Methods as instance methods ?

For example, C# code (Anonymous Method):

internal class AnonymousMethod_Example
{
    public void MyFunc(string[] args)
    {
        var x = 5;
        Action act = delegate ()
        {
            Console.WriteLine(x);
        };
        act();
    }
}

Will produce IL Code (Anonymous Method) similar to:

.class private auto ansi beforefieldinit AnonymousMethod_Example
{
    .class nested private auto ansi sealed beforefieldinit '<>c__DisplayClass0_0'
    {
        .field public int32 x

        .method assembly hidebysig instance void '<MyFunc>b__0' () cil managed 
        {
            ...
            AnonymousMethod_Example/'<>c__DisplayClass0_0'::x
            call void [mscorlib]System.Console::WriteLine(int32)
            ...
        }
        ...
    }
...

While this, C# code (Local Function):

internal class LocalFunction_Example
{
    public void MyFunc(string[] args)
    {
        var x = 5;
        void DoIt()
        {
            Console.WriteLine(x);
        };
        DoIt();
    }
}

Will generate IL Code (Local Function) similar to:

.class private auto ansi beforefieldinit LocalFunction_Example
{
    .class nested private auto ansi sealed beforefieldinit '<>c__DisplayClass0_0' extends [mscorlib]System.ValueType
    {
        .field public int32 x
    }

    .method public hidebysig instance void MyFunc(string[] args) cil managed 
    {
        ...
        ldc.i4.5
        stfld int32 LocalFunction_Example/'<>c__DisplayClass1_0'::x
        ...
        call void LocalFunction_Example::'<MyFunc>g__DoIt1_0'(valuetype LocalFunction_Example/'<>c__DisplayClass1_0'&)
    }

    .method assembly hidebysig static void '<MyFunc>g__DoIt0_0'(valuetype LocalFunction_Example/'<>c__DisplayClass0_0'& '') cil managed 
    {
        ...
        LocalFunction_Example/'<>c__DisplayClass0_0'::x
        call void [mscorlib]System.Console::WriteLine(int32)
         ...
    }
}

Note that DoIt function has turned into a static function in the same class as its parent function. Also the enclosed variable x has turned into a field in a nested struct (not nested class as in the Anonymous Method example).