Perhaps prodigies are defined differently in different fields, but in music I think the convention is someone who performs at a level that gifted adults can reach but not world-class. For example, a 10-year-old who can play piano as well as a reasonably good piano teacher, but not as well as a world-class performer. In this sense I think Tao is not a prodigy (when he was 10 he probably wasn't as good as a decent math professor). But nonetheless he was one of the best of his generation at that time so maybe he is. A small number of 8-, 9- or 10-year-old chess players are rated higher than very advanced adult players and these are probably prodigies. But really these are semantics in a sense. A parent needs to nurture the kid as best as possible no matter what label fits.
Andrew Solomon's chapter on music prodigies, though, has very detailed account of how the music prodigies are brought up. Some are forced, but many are not. Many do love playing instruments hours a day. And some didn't even need to practice like crazy to achieve the "prodigy" status (for example, Joshua Bell). Some parents pushed the kids; other simply gave them what they asked for. Of those children who were pushed by their parents, some resent it and have major issues with their parents even after they grow up; others felt grateful that their parents did push them. So it comes in all varieties, as should be expected.
