I finally discovered the kind of toy that my male blackcap caique prefers! It's a cheap and very simply toy: a twine threaded through holes in smallish yucca medallions (http://www.abirdtoy.com/Small-Bird-Toys_c_17-2.html). It's easy to find and relatively inexpensive - and the only reason why I say 'relatively' is that, although it is cheap, he goes through one from beginning to smithereens in less than two days so I need to buy a number of them every week.
Yes, my Codee GCC likes them, too - as do the Indians, the budgies (they LOVE all yucca toys) and Keku quaker. I guess the fact that they are small and soft provides a strong appeal to the small species.