"I'm trained in American style. I've been working with a studio that does International style. My question is this, is there a difference in Rhythm dance frame between the two?"
Possibly, depending on who is teaching.
"It seems as though at the new studio, they use a smooth ballroom frame for every dance they do, including all latin dances. Is this correct for International?"
The proper frame for basic closed hold actions in latin would have a lot of similarity to a ballroom frame, but it's not identical to one. There's little offset, the lady's hand would be more around the man's arm, etc.
This may not be an area that gets a lot of attention because a full frame is not seen that much at advanced levels. When couples are close enough together to permit one, they are as likely to be in some alternate hold of personal preference. So a strict frame is mostly a teaching concept.