I believe the definition of a prodigy is a child, usually under the age of 10, having the level of ability as an adult.
This is roughly what I have read about.
There is a very interesting chapter in Andrew Solomon's book Far From the Tree that discusses musical prodigies. Those are the kids who, before reaching adolescence, performed at the level of a reasonably accomplished adult. Talent is definitely a huge part of it, but it's not just talent, it's talent well developed when they were still 8-, 9- or 10-year-olds. So there is usually a lot of effort involved.
However, the caution is that many of these prodigies don't grow up to be world-class musicians. There are many reasons, some burn out, some simply plateau very early, some resent the entire idea of being a prodigy and rebel, etc., etc.
On the other hand, studies of world-class achievers (in music, math, science, etc) in the book Developing Talent in Young People have shown that these people were almost all not prodigies when they were children. They obviously had some talent but didn't stand out that much. What was different for them was the passion and the persistence.
