Below is a little chart I made. I hope this clears up some confusion as to how Ballroom is divided into styles and dances.
Notice how American and International both have the same names for certain dances. For example, Foxtrot is a dance in Standard and Smooth.
While this happens in several places, this by no means indicates that they are the same dance, although they are quite similar in most cases. The problem is, some genius American decided to name half their American Style dances after their International counterparts, and this proves to be quite confusing for some students, and understandably so!
Enjoy, and I hope you learn something!