Alternative Y combinator definition

I've spent some time wrapping my head around the Y combinator lately, and I've found that it is usually defined (more or less) as follows (this is in C#, but the language of choice isn't important):

public delegate TResult SelfApplicable<TResult>(SelfApplicable<TResult> r);

public static TResult U<TResult>(SelfApplicable<TResult> r)
{
    return r(r);
}

public static Func<TArg1, TReturn> Y<TArg1, TReturn>(Func<Func<TArg1, TReturn>, Func<TArg1, TReturn>> f)
{
    return U<Func<TArg1, TReturn>>(r => arg1 => f(U(r))(arg1));
}

While that's perfectly functional (pun intended), it would seem that my definition is much simpler:

public static Func<TArg1, TReturn> Y<TArg1, TReturn>(Func<Func<TArg1, TReturn>, Func<TArg1, TReturn>> f)
{
    return f(n => Y(f)(n));
}

Is there any reason why the latter definition is not as common (I have yet to find it on the net)? Would it perhaps have something to do with defining Y in terms of itself?