IME training horses, I agree with the majority on it being a nature/genetic thing. Just as people, horses can be born with more go than woah (extroverted - always wanting to move their feet) versus more woah than go (introverted - prefer to move their minds rather than their feet). I’ve currently got a Dutch Warmblood cross, a Thoroughbred, and a Paint who are introverts. The Warmblood gelding is probably the ‘laziest’, but you wouldn’t know when I ride him, as I have learned how to build impulsion in him and earn his respect. The Thoroughbred never made it on the track because he was so slow and relaxed!! Don’t get me wrong, he was bred to run and does enjoy a good romp, but his desire is not so much to move his feet. Conversely, I’ve also got another Thoroughbred (successful on the track) and also a Quarab (running-bred on the QH side) who honestly cannot keep their feet still for a moment - those two love to run, with a passion!! In earning their trust, offering appropriate leadership, and teaching them relaxation though, they can be as soft and easy to ride as the Warmblood. I definitely do prefer the more forward rides though, typically, as I feel I have more to shape and work with (my new Warmblood filly is an extrovert - exactly what I was looking for!).
Training will enhance certain qualities: for example, the dull horse can be made light, responsive, and quick on its feet with good training and the extrovert can be created into a relaxed partner who has no problem standing for long periods of time (when I first brought home my extroverted TB, he had no idea what standing still was all about, lol, yet now he’ll stand quietly - relaxed - for as long as necessary). Proper training 'balances’out/‘centers’ the different personalities so that it is (relatively) difficult to distinguish extrovert from introvert: impulsion is built in the more-woah-than-go horse, and relaxation, rhythm, and looseness are built in the more-go-than-woah horse. Poor training will only exacerbate a horse’s ‘nature’. For example, riding a ‘dull’ (introverted) horse ‘heavy’ will only result in a ‘duller’ horse who blocks out the rider’s aids, while riding a high-energy (extroverted) horse ‘heavy’ will result in a horse who is too forward and is fearful/reactive. My job as a trainer is to make the forwardaholic relaxed and the lazy horse have impulsion, so that someone on the ground cannot tell the difference between my rides.
Physical conditions (such as being out chiropractically, bodysore, etc etc), disciplines (ie. doing what the horse loves or giving them a specific job), etc, can also either build or detract impulsion from a horse and thus create more woah or go in a horse too, obviously, but their personality underneath usually remains the same.
Kind of on a different-ish note, I don’t really like the terms ‘push’ or ‘pull’ either, because a rider should be doing neither. You should never be constantly pushing or nagging a horse, and neither should you be pulling on a horse.