There is something called "classical ballet training" which is a set of techniques that are used methodically in the process of building a "classically-trained" ballet dancer. It would be expected that a classically-trained ballet dancer would be able to perform a series of movements with a certain degree of strength, and those movements would be movements that a person without "classical training" would not be able to do. What would be parallel in music to this? Being able to play scales and arpeggios in all keys? Is that what playing music is all about?
In the case of performing musicians, it seems that the main reason for someone to call him or herself "classically-trained" is to make a contrast with someone who has only learned to play his or her instrument through playing music that is not "classical," but it is often used as a way of giving more cachet to they idea of having taken (and paid for) private lessons, or knowing how to read music. The term "training" implies the loftiest idea of private study: training at some kind of "classical" academy, at the hands of a "trainer," who demands serious applied discipline. Some teachers do demand discipline, but many don't, or they don't know how to teach students to develop self discipline.
Many of the good musicians I know have developed their techniques after a period of study, and usually it was study with a good teacher. Some have developed excellent technique in spite of having a teacher who was not very good. Some have done it without much help from a teacher at all. It is rare that a violin teacher would ever think of him or herself as a "trainer." Even Suzuki teachers, uniform as they try to be in their methods and materials, offer different perspectives, different senses of sound, slightly different ways of holding the violin and the bow (there is no right way because everyone's body is different), and a different sense from one another concerning the whys and wherefores of playing music.
My own violin "training" is not training at all. I practice. I learn what I can from whoever I can. I even learn from people I would never have the chance to meet, and I even learn from people who are no longer alive. I learn from people who don't even know that I am learning something from them. There is also a lot of great musicianship and technical strength to be found in the playing and singing of people who have never had any interest in playing "classical" music or "getting" "classical training."
I apply my own methods of practice to material that I choose to use to develop technique. I apply my own interpretations to the music that I play. I could never in a million years call myself a "classically-trained" violinist or violist, and my students could never in a million years think of what they are getting from me in lessons as "classical training." Sure, I encourage them to read music, have strong hand positions, play in tune, and think about the music they are playing as vehicles for expression, but it doesn't come from any "training."
Wouldn't it be nice to either eliminate the "I'm a classically-trained" this or that and substitute it with a phrase that is a bit more realistic?