I agree with you to a point. The geniuses we remember do generally have a certain "gift" and then, through rigorous training/practice/improvement reach heights that most people cannot attain. On the other hand, some "gifted" geniuses do not apply themselves in order to rise above the masses. They enjoy their "gift" to the extent they use it, but do not attempt progress to the level they could have if they had applied themselves.
I only say this because I feel I have a certain "gift" that I enjoy when I partake of it. My "gift" was recognized and, to a certain extent, celebrated when I was a child. Had I applied myself in pursuing this "gift", there is a possibility I could have gained some notoriety (nothing is certain in this world). However, that was not where my interests were at the time. Looking back, I sometimes wonder how my life would have been different has I applied myself in that area. However, I still enjoy my "gift" when I use it.
So, I guess my question is, does "genius" require notoriety? Does application and/or advancement of your "gift" factor into being a genius? Or is it just being better than just about everybody else at whatever your talent is?