How could I forget, Ramanujan is also a good example of a genius with little university experience. He also possessed abilities that almost no person has. His abilities and use of these make him a genius (he's one of the handful per century).
Yes, I think we have similar intuitive definitions of 'genius'. I think the word 'genius' is probably thrown around a bit too loosely. To me it means someone with exceptional (e.g. 1 in 10 million) cognitive abilities. Of course, the specific cognitive abilities in question could vary. It would be hard to have more than an intuitive definition of the word 'genius'.
But I don't think Bill Gates would qualify by my definition. He is talented, and smart, but I wouldn't be surprised if there were loads of people with similar abilities in silicon valley.