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All pokemon have at the very least a low level psychic presence. This is reflected in a few ways, like how they can understand humans (especially trainer commands) and communicate with other pokemon despite having no shared language. It also shows up in a few other ways; in particular in the anime, you notice how one moment a Hyper Beam might level a building but the next a Flamethrower right to the face of a human only leaves some soot? Most pokemon are good hearted and not killers at the core, and generally (subconsciously perhaps) pull their punches so to speak when it'll do permanent, severe harm to a person or other pokemon. Thats why they can use all these crazy attacks in battle without killing each other. There are exceptions, but they're rare. This applies more to the anime though, as pokedex entries in the games seem to paint some pokemon as being more vicious.
Building on the psychic presence thing, when the term "strong trainer" gets bandied about its not just referring to someone with a good team, trainers with a strong force of will and personality actually empower their pokemon to perform above and beyond what they'd do alone. If you took say, Cynthia's Garchomp and gave it to an average trainer, it would underperform even if given exactly the same instructions in a battle, because Cynthia has a stronger will than the average bear. The same goes for most significant trainers, including the MC's of the games.
