Um, ivory is from elephants. Rhino horn isn't ivory.

no, it's also from rhino horns.

No. Rhino horns are made of keratin (like hair or fingernails). It is valuable, but is often used ground into a "medicine", not for ornamentation.
Ivory is composed of dentin, like teeth. Sources of ivory include elephant tusks, walrus tusks, and narwhal tusks.

When I was a kid, I liked ivory. Now I know better. Stop producing ivory NOW!!!

Strictly speaking, ivory is made from the elephant's tusk. So it's not ivory until the tusk is turned into something else. Perhaps that's splitting hairs, but I don't think so.