Rykas, my knee-savers! 
I use my Ryka aerobic shoes for both cardio and weights. A lot of the differences between the shoes for various activities lie in where and how much cushioning and support they have, based on where the impact and stresses are for that activity. ( More cushioning in the heel for running shoes vs. more cushioning in the toe for aerobic shoes, for example ) Since there really isn't any IMPACT per se in weight training, I think any shoe that is comfortable and has sufficient tread so your stance doesn't slide around would be fine. I don't see why crosstrainers would be necessary for weight training.
I have tried other brands, but nothing gave my foot the support it needed so desparately as Rykas. I prefer a mid to high top, rather than the low cut, because I need the ankle support ever since I sprained my right ankle in a fall 2 years ago, but that's an individual preference. Also, if you're trying to decide between 2 different aerobics shoes, I have found that having a somewhat more flexible sole to be more comfortable in stepping, since there is so much toe ball heel as you go up and down on the step.
A note of caution: I have, on occasion, been told by uniformed ( or perhaps sales hungry ) sales people in stores that cross trainers are just as good for aerobics as aerobics shoes. This is usually when they are out of my size in aerobic shoes.
This is NOT the case. For the kind of high impact you're doing in Cathe's videos, it's VERY important to have shoes designed for this activity.
I hope this helps! Have a great night!
Erin
