Big Name Trainer to me suggests name recognition outside of the immediate town trainer is in, at least to people in the discipline. Perhaps someone that does clinics or travels.
It is not to me an endorsement, it doesn’t mean excellent horseman or trainer I would recommend or any evaluation of this trainer over other comparable ones.
It is also a very loose term because obviously anyone can say well, I never heard of so-and-so. It may be a term that is often used generally but is less precise in application to individuals.
I am sure there is lots of room for debate over a lot of regional names if you wanted to debate whether a BNT has name recognition internationally or nationally or regionally, and whether my not knowing someones name is a sign that they are indeed obscure, or a sign that I am ignorant 
It isn’t really meant to be precise. It’s kind of tongue in cheek too.