Influence - the power or capacity of causing an effect in indirect or intangible ways.
That's what Mirriam-Webster says, but that's a pretty loose definition, huh?
I believe this is a tough issue. Based upon the above-mentioned definition, I could say that I'm influenced by the sound the clicking of my keyboard makes, the rhythmic thump of my tires as I drive down a cobblestone road, etc etc. In fact, many of the most "influential" musicians have stated that they hear music in everything in everyday life. Nowadays, we tend to think of influences as "the main type of music an individual listens to, wherein patterns, concepts, or ideas from that musician can easily be directly traced back to said genre." Okay, that's a quick off-the-top-of-my-head definition.
There are several things I tend to agree with that have been said. I think jazz is a very honed skill that relies more heavily upon "musical knowledge" than some other genres. By "musical knowledge" you could say, harmony, theory, meter, whatever. I know I might get slammed for that statement, but let's be realistic, a 2 month old bassist can easily be taught to thump out 8th note roots over a two chord pattern, than to understand the intricacies of modal jazz. What I want to make clear, is that I'm definitely not saying that means one type of music is "better" than another, just that it's more involved or complex. Hell, my cell phone is a more complex instrument than my eyeglasses, but...oh what's my point?
The point is, I do really think that a lot of people "name drop" in regards to jazz, and saying they've played jazz, been influenced by jazz, or listen to jazz. Someone here said it best, there is a very clear difference between a jazz musician and someone who plays jazz. A jazz musician is someone who really lives and breathes the music, they understand the subtleties, history, and concepts behind the genre. Obviously all music has history, but there are not really many styles that can claim as rich and important a history as jazz. Jazz's history is not just that of music, but of celebration, oppression, redemption, acceptance, and more obstacles than I could imagine.
I don't know about one specific individual, but personally, I also tend to take comments regarding one's influence by jazz with many grains of salt.