The trouble with ponies is often that they are too little for an adult rider to ride and train them, and so all their training has been done by little kids who yank them around. If they are a little rank it is because no one puts the time into training them that they would a full sized horse, they think they're more harmless because they're small.
That makes sense, about adult vs. child training. It did seem to be a power play between the humans and the pony - the pony looked for new ways to win the day.
Technically, it's a "pony" if it's anything under about 14 hands. Harry's already described it as "a small horse," so I think it's probably on the upper range of "pony."
That said, the rest of your points are right on the money.