What Really is a Musical Genius?

The definition for 'genius' reads as such: 

1. exceptional intellectual or creative power or other natural ability.  
"she was a teacher of genius" 

2. a person who is exceptionally intelligent or creative, either generally or in some particular respect.  
"one of the great musical geniuses of the 20th century" 

I'm pretty sure most people would define a genius as "really smart" and regard that person pretty high in our society. I guess that's pretty close to the definition. However, there's a few words that stick out to me in these two definitions: "natural ability" and "in some particular respect". 

The first definition states "natural ability". So they were born this way? They didn't have to do anything to nurture this ability? Does that make them smart? I really think this label gets thrown around too much. Most of the highly regarded musicians and composers would scoff at the idea that they didn't work hard to achieve their status. Some might've worked even harder without getting any notoriety for it. But the geniuses just wrote music that they heard in their head. One could say, genius is natural and automatic therefore it is not an achievement. 

Some contemporary musicians and composers who have studied at some of the most prestigious colleges have regarded the rapping rhythms of hip-hop music as genius. Was there a well thought-out procedure to procure these intricate rhythms? I would agree with this statement pertaining to some rap artists. But what the hell is a musical genius then? Somebody that is naturally creative at the right time in pop culture? 

Geniuses are rare. Having six toes is rare too! 

I bring this up because a lot of people talk about their favorite musicians as "a musical genius". We're talking about music here, everything is subjective! If nine people told me that Frank Zappa was a musical genius, the tenth person would tell me his music is terrible. Sure, he's creative, but is he genius? Years back, I once argued with a friend that Trent Reznor has been regarded as a musical genius and therefore I should check him out. He thought that it was a waste of his time because he didn't hear any genius in his music. Sure, he's influential, but is he genius? 

To me, a "musical genius" is an oxymoron. How can a personal taste qualify as being intellectual or "smart"? What do you think? Is there such thing? And if so, who are some of these musical geniuses? What did they specifically do to show high intellect or creativity?