I'm going to provide a contrasting viewpoint here, while still acknowledging that Master of Design's approach can work.
Luis, although it's too late now, I think that this outcome would have been better if your approach was much more relaxed. Instead of presenting your (factually well-established but) rather aggressive case, you could have simply started by making some beans in the kitchen, and adding them to your meals. No mention of vegetarianism was ever necessary. You could have just told your family that you tried some beans at school, and you'd like to have some at home. Over time, you could have eaten more beans and less meat. Your family might have even joined you.
Luis, your family's argument is not about facts. It is about your parents' desire to raise you a certain way, determined according to their sensibilities. They believe that they know better than you, and (considering that you're only 15 years old) this is probably true in many respects. In regards to vegetarianism, they are incorrect. They may be afraid of losing their ability to guide you in other, more critical ways. They may fear that your vegetarianism will lead to other choices - ones that they disagree with (progressive politics, for example).
Luis, my advice is to relax, and accept your parents' compromise to eat less meat and more beans/lentils. Relax - it takes time for people to change. Over time, your parents may also relax their fears about vegetarianism. It won't be long before you are living independently, and then you can do what you wish. I didn't become a vegan until I was 22 years old. I’m now 51.