Why do muslim extremist say they are muslim when muslim do not claim them?

I have read some of these questions and answers, that make alot of sence but there was one that made me think of this and i am not bashing anyone that is a muslim or any other religion all i want to know is why muslim extremist say they are killing in gods name (allah) but other muslims say that they have no religion or they are mis-interpreting it. Someone on a answer wrote that all muslims should participate in jihad and kill the infadels how can you mis-interpret that? i mean if you dont believe in islam you should die! right? i know that othere muslims say that, that is wrong and those souls or people should be given a safe page and prayed for, right? so i guess what i am asking is why everyone says that there is 2 different types of muslims the extremist and the right muslim that does not bother blowing anyone els up, if it cleary states in the qu,ran that all muslims should participate in the jihad?