Question

When a cutting from a plant sprouts roots and yields a new, independent plant, what process has occurred?

Answer 1

This is "vegetative reproduction"

Explanation:

from Wiki
Vegetative reproduction
(vegetative propagation, vegetative cloning )
is
• any form of asexual reproduction occurring in plants
• in which a new plant grows from a fragment of the parent plant
• or grows from a specialized reproductive structure
(such as a stolon, rhizome, tuber, corm, or bulb)
https://en.wikipedia.org/wiki/Vegetative_reproduction

Higher plants usually reproduce by sexual reproduction, where pollen from one flower fertilizes the ovum of a different flower, which produces seeds.
But many plants also have the ability to grow right from cuttings.