Core is the whole thing less pads.
On earlier Jags (and other Brit Marques) the seal is on the inner part of the piston, wiped up the bore as pads wear. The "dust seal" between the visible end of the piston and the caliper housing is all that keeps water out.
Calipers rust badly when water gets to the cast iron, and then, as the pads wear, the seal moves into...rust. And they leak.
Newer designs puts the seal in the caliper housing proper, at the closest end, piston rides on seal, everything behind the seal is in brake fluid.
If that's how yours are (and since you see rebuilts for sale, I think they are), your brake shop should be able to rebuild yours.
If the pistons are good, just a few bucks for a kit, big bore hone knocks any rust out of the bore, scrape the seal groove out to make sure it's clean, and put it together.
Once off, about 20 minutes per caliper bore is more than enough time.
Dave