A religion or religious sect generally considered to be extremist or false, with its followers often living in an unconventional manner under the guidance of an authoritarian, charismatic leader.
The followers of such a religion or sect. A system or community of religious worship and ritual. The formal means of expressing religious reverence; religious ceremony and ritual.
Allah is the Arabic word for God.
Mohammed always stated that he was a messenger of God and it is westernized interpretations that began to say call followers of Mohammed Mohammedans. Muslims take exception to that because it gives the incorrect impression that Muslims worshipped Mohammed. They do not. They worship God and accept Mohammed as his prophet and messenger.
Mohammed believed he was the last prophet of God (Allah), the same God of Christianity. He was upset that his country men were wild and lawless, believing in idols.
I can find other areas of dispute but have decided not to finish reading because I think what you have chosen to print here is a biased and possibly inaccurate and unfair interpretation. If you read the World Book Enclyclopedia, for instance, there you will learn that Mohammed hated war and taught against violence except for self defense. He tried desparately to spread his beliefs most of which were centered around goodness and fairness within the family and assistance of the poor.
The Jihad was something that Abu Bakr and his successors began. I think we need to be careful in studying Islam that we do not begin with a source of information that is biased by their own religious beliefs.