I could talk A LOT on this subject.
I don't think that being willing to die for your views or beliefs is extremist. At all. I think it's very typical of someone who is dedicated and secure in their beliefs and has the ethical standpoint that you don't waiver in the face of persecution. Whether religious, moral, political.
I also don't know that I agree with extremists not viewing themselves as extremists. I think in a real case of extremism (willingness to kill for your beliefs, a strong isolation from General society based on unpopular views - and I don't mean not being in the majority, I mean only a handful of people in the country or world share your views), people know their views are extreme.
Examples of people I would consider extremists: terrorists, people who believe we could benefit from genocide, people who believe homosexual relationships should be punishable by law, people who believe those who eat meat are some kind of sociopathic murders (not just a general meat-is-murder idea).
I have a lot of friends who are vegan, some maybe radically so. They share information on veganism and aside from believing in health benefits or animal rights, they believe a widespread vegan movement would benefit the economy and environment. They actively pursue educating people on and spreading their beliefs. But I wouldn't call them extremists, as so many people do believe in them, they do have rational reason to believe what they do, and they don't harbor animosity towards those who disagree.
I know one person, who is what I would consider an extremist vegan. She regularly gets into heated arguments over it, experiences a lot of turmoil over people who don't follow her ways, does not allow animal products in her home whatsoever, calls people who eat meat "carnies" as a kind of slur, and truly believes if you eat meat you are an active participant in animal abuse. I think she knows her views are extreme, although I can try to ask if she views herself as an extemist.