i agree with OP.
to me, a pokemon master is someone who has the drive to become stronger. with this, every other thing falls into place.
a pokemon master will have at least a grasp on the different fields of pokemon study - evolution, stones, types and breeding - and eventually the other more specific aspects that affect the growth of a pokemon strength-wise and not exactly level-wise - ev training, stats, items, characteristics, berries, abilities, weather-effects, etc.
with this he will be able to find and predict the weaknesses of a certain pokemon and turn those weaknesses into advantages that will make him have the upper hand in pokemon battles.
a pokemon master is able to make a jigglypuff defeat a gyarados. in other words, he doesn't solely rely on levels to determine whether he can win a battle or not. he defeats all his opponents using the wit and the knowledge he acquired from his journey. and by wit and knowledge, i mean knowing the right movesets, items to hold, turning a type disadvantage into an advantage, etc.
TL;DR pokemon trainers who wish to be masters should literally master the study of pokemon. if he does that he'll be able to turn literally any pokemon that falls under him into gold. he becomes invincible after that.
that's what makes a trainer a "master" in my vocabulary.