Yes, a flexitarian is a part-time veg but still sometimes eats animal products. A vegan does not consume or use any animal products for ethical reasons. If you consume a vegan diet but still buy animal products to wear, clean with, etc, then you consume a plant-based diet but are not technically vegan.

The problem with saying plant-based is that it implies "based on plants" with no rigid definition. Some people think it means "vegetarian but not vegan" or just "mostly plants." I have even seen products that contain animal products that I am allergic to marketed as "plant-based," so it's just not a good term for me.