I am following the SWSY weight resistance program, and I am also enrolled in a "step" class. While the class is mainly aerobic exercise, we also spend at least 10 minutes a session with weights, bands, or body bars doing weight resistance type exercises. My question is: should I count this as weight training, and give myself rest days around the classes?

That depends on the intensity of the strength training. If you're working the muscles to fatigue - to a point where you couldn't continue in good form - then they need a day of rest to recover.