If a piano player is 'trained' or experienced in a particular area -- then they can indeed have the option/choice to 'train' in another area at some time - such as 'classical' - in which case the person will be/become 'classically' trained too.
So basically - they can go one way or another - or even both ways at the same time. As we know - there is a lot of variety in piano players, because people not only have different 'training' - but different in their own potentials, ways of thinking, feelings, understanding etc. It all depends on goal/aim etc. And importantly - one 'style' of music isn't better than another style. So if somebody can't play every bit of music that was ever written or performed, then it doesn't mean that they're 'lesser' than somebody else - even the 'world class' professionals ------ because everybody is different. A 'world class' professional might not be able to 'think' in the same way as the 'other' person -- and hence the a world class professional is not necessarily 'better' than others at the piano -- and generating particular music. And as mentioned - 'classical' music is not 'better' than other types. To me - they're all amazing, excellent, outstanding. I know that others have different opinions. But that's ok. It's good to see how people out there 'think'.