Both methods work, but they trade off different results. Starting eggs in cold water and bringing the pot to a boil heats the white more gradually, which tends to produce tender whites and evenly set yolks—but those slowly heated proteins can bond more tightly to the shell membrane, making peeling harder.
Starting eggs in already boiling water (or hot steam) rapidly denatures the outermost white proteins so they shrink and pull away from the membrane, which is why peelers usually find hot-start eggs much easier to shell. America’s Test Kitchen’s testing found cold-start and baked eggs tougher to peel, while boiling- or steam-start eggs released far more cleanly.
For easy peeling, drop refrigerated eggs into boiling water (or steam), cook to your preferred doneness, then chill in an ice bath. Use a cold start when tender whites matter more than a perfect peel. Source: America’s Test Kitchen — Easy-Peel Hard-Cooked Eggs.