A trained musician, in my view, is a superb craftsman. This means that one has a firm hold on harmony and counterpoint, a good understanding of musical form, and an ability to apply the knowledge of these subjects to the performance and creation of music. As I'm sure you've guessed, there are very few well-trained musicians in my opinion.
I wasn't going to mention the obvious, but perhaps I had better. A trained musician, in my view, must have a very refined ear.