You're correct about total internal reflection. The light in the core must move more slowly than in the cladding. The majority of the light remains in the core

You're correct about total internal reflection. The light in the core must move more slowly than in the cladding. The majority of the light remains in the core though some travels in the cladding as well. A higher index of refraction means lower speed of light so you would expect to see the core have a higher IoR than the cladding.