[Jim Noble]

Since the Constitution never actually defines the term "natural born citizen," and thus is falls to Congress to define that term

No, it doesn't. Congress has no such power.

[Yashcheritsiy]

It would seem to be implied in the fact that Congress gets to define, per their power to establish uniform rules of naturalization, what isn't a natural born citizen. The question of implied powers was pretty much settled from the very beginning, so yes, Congress does get to define it.