Beautiful Jefta! I am sooooooo jealous of nicely she gives you focus - that's been one of the most difficult things for me to master with my dogs.
Formal Obedience with the AKC looks very similar (in nice form, anyways). Rally Obedience looks muuuuuch different, since there is no limit to how many commands (either hand or voice) that you are allowed to give
. It looks, in many cases, with Rally Obedience, like the dog is charge of the owner.
Rally is a good "gateway" to AKC obedience for newer handlers though, and if you are serious, and make a point to have a nice presentation and well trained dog, you can score highly, often.
We also have IPO obedience training in both countries though - through an organization called the DVG.
http://dvgamerica.com/
This training is more strict, and much more similar to what you see in Europe.