How is New Wash different from sulfate-free shampoos like Prose or Olaplex?

Sulfate-free is not detergent-free. When consumer awareness of sodium lauryl sulfate grew, the industry replaced it with other molecules - cocamidopropyl betaine, sodium cocoyl isethionate, decyl glucoside, and others. These are detergents with different names. They still foam. Because they foam, they still strip. Changing the molecule did not change the mechanism. The INCI naming system makes this easy to miss. "Sodium Cocoyl Isethionate" contains "cocoyl" because its fatty acid chain originates from coconut oil - but the finished molecule is industrial. Natural origin of a precursor does not change what the finished molecule does. Prose contains Sodium Lauroyl Methyl Isethionate - a detergent. It foams. It strips. New Wash contains no detergents at all - not sulfate-based, not sulfate-free alternatives. This is why "detergent-free" is a categorically different claim from "sulfate-free." Most free-from claims remove one ingredient while leaving the mechanism intact. Detergent-free removes the entire class of molecules responsible for the stripping.