I've opened both the Radio Shack and generic induction pickups to repair broken wires. The pickup's connecting wire is stranded but it still breaks if it's flexed enough. That usually happens right at the hole where the cable exits the pickup body. There is a strain relief (i.e., a simple overhand knot in the cable) inside the pickup body to keep the cable from being outright pulled off the coil connections but there is no strain relief for the cable outside the pickup. I used some RTV gasket maker compound to make an unattractive but effective strain relief at that spot and I haven't had a broken wire so far.
Using dial calipers, I get a diameter of 0.191 in (4.85 mm) and a length of 0.54-0.55 in (~14 mm) for both the Radio Shack and generic induction pickup cores. I see that site you linked to has medium permeability ferrite rods that are a little smaller in diameter, 0.175 in., and 60% longer, 0.875 in., than the steel cores. A bit of paper or tape wrapped around those ferrite rods should make for a better fit.
I *assumed* the steel cores were made of mild steel based on their color and the price of the pickups. It would be nice if the Radio Shack core turned out to be silicon steel because that material has a much higher magnetic permeability than mild steel and that's why the Radio Shack pickup has a stronger signal.
