Vegetarian -- Anoun. A person who on principle abstains from animal food; esp. one who avoids meat but will consume dairy produce and eggs and sometimes also fish. [...] M19.
Vegan -- Anoun. A person who abstains from all food of animal origin, and avoids the use of animal products in other forms, e.g. the wearing of silk or wool. M20
Thus a person who eats a whole-food, plant-based diet may not be a vegan and a person who eats a vegan diet may not be someone who eats a whole-food, plant-based diet nor even a vegan.