Should we correct Buddhist wrong believes or respect it?
Those rules you have mentioned are only for monks. They do not apply to lay people. Lay people can consume any type of meat even if the animal is killed for them, as long as they don't break the 1st precept. 

Yes, you can try to explain to people who look down on you, saying that being vegetarian is merely a food preference in Theravada Buddhism. Mahayana Buddhism seem to recommend a vegetarian diet by default. But that is not an excuse to look down on someone. You might also come across some confused Theravada Buddhists who follow vegetarianism like a religion, even putting it above what's said in the Suttas. Such practices usually fall under Silabbata-paramsa. One gives up on that when one becomes Sotapanna. Trying to correct them might push them away from the Dhamma. You can always practice Upekkha alongside Metta in such cases.