Should we correct Buddhist wrong believes or respect it? Allow me to give the exemple of meat eating for this question: In the suttas we can clearly see (many times) that eating meat is accepted in Buddhism under a few conditions. We cannot kill, ask someone to kill or hear/see the animak being killed.

In addition some meats should not me eaten (human flesh, elephants etc...), but most of the meat we eat today is clearly accepted in the suttas, still there are people that look down on you if you eat meat, mixing personal views with Buddhism.

There is a sort of "main stream" Buddhism with things clearly not coming from the suttas, in this case, should we correct them? Or should we respect other peoples view?
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.