It does depend on the individual horse for sure....but the three that I have met, I know...not very many at all, but all of them have been dominant, assertive horses. I met my horse when he was 10months old, and his brother, who was 2. And then another I met was at the barn I had my horse at, and they all had very similar personalities, in that they were very particular in the people that they liked, if they didnt like you, you knew it, and if they did, you knew it.
My horse will do quite a bit for me, but with some other people, he'll act like he has no clue, or has never learned anything before. These three horses could just happen to be similar due to their bloodlines though.....but like you said, def depends on the individual horse!