English has traditionally divided the colors of the rainbow into seven different colors ever since Isaac Newton described it as being that way in his groundbreaking treatises on optics. It was his work that divided the color pattern made by splitting white light with a prism, the same gradient made visible when white light is refracted through raindrops, into Red, Orange, Yellow, Green, Blue, Indigo, and Violet segments, and these are the terms that have long been taught to Anglophone children as being the names and order of the colors of the rainbow. Note, interestingly enough, the presence of a prism like Newton's in the work at hand.
When Nakatani mentions differences in color terms across languages, she's probably referring most directly to the work of Brent Berlin and Paul Kay, who performed a wide survey of color terms across different language groups. They concluded that color terms are generally present in languages according to a certain hierarchy; that is, color terms tend to emerge in languages in a particular order, and languages with fewer color terms tend to group certain shades together that languages with more color terms tend to differentiate.