Kate, you can definitely use any brand of chipboard cover with the Cinch, but not with the BIA. One of my biggest problems with my BIA (and the main reason I got rid of it) was that I felt I was really limited to ONLY using the Zutter brand chipboard with it, and then sometimes once I covered it with paper that had any thickness to it (and there is a HUGE variation in thickness of cardstock and scrapbook paper out there) I would sometimes have a major problem getting it into the machine to bind it - like it was too thick once the paper was on there to fit it in, and often I would have to force it in and then it would get all messed up. I could have punched my chipboard and paper separately and then glued them together, but I felt that this defeated the purpose of having an at-home binding machine.
Personally, I felt that the Zutter chipboard was too thin for my liking - some projects I wanted to use a thicker chipboard for my album or book cover, but that wasn't an option with the BIA as it wouldn't fit into the machine.
I've been really happy with that with the Cinch - It doesn't seem to matter how thick the chipboard is, it'll fit and it'll punch through with no problem. After the first time I used my Cinch, I got rid of my BIA because I knew I'd never use it again.
I realize this issue with the cover thickness is getting away from your original questions though. About the oings, I haven't had much luck finding that variety of colors for the Zutter like Karen found - I just found them in white and black, but maybe I wasn't looking in the right stores online? Again, that would probably just go back to personal preference as I don't see myself using the colored rings - my personal style would probably only have me using the silver, black and white. But I can say for certain that the Cinch rings are MUCH sturdier and higher quality than the Zutter rings - so I don't mind paying the slightly higher price (although for me here on Maui, the price is really almost the same once it's divided down to individual piece of wire, even if I bought the Zutter wires in bulk).
I would definitely try to go somewhere and look at the binding wires and compare them in person to see what you think before making a decision - you might find that you have a personal preference for one vs. the other. I have a friend who strongly prefers the Zutter wires despite everything I've described above, so you really just never know!