Taken from the Concise Oxford English Dictionary:
- Mussulman – plural Mussulmans or Mussulmen – archaic term for Muslim. From Persian musulmān, from muslim. [So it is valid, just old.]
- Moslem – variant spelling of Muslim. [So that is also valid.]
- Muslim – a follower of Islam. From Arabic, active participle of ‘aslama. [So if in doubt, use this.]
- Muhammadan or Mohammedan – not favoured by Muslims and should be avoided.
- Islam – from Arabic ‘islām meaning submission, from ‘aslama meaning ‘submit (to God)’.
- Islamism – Islamic militancy or fundamentalism.
- Islamist (or Islamicist) – derivative term from Islamism, so militant Islamism or Islamic fundamentalist.
So, IS / ISIS / ISIL could be called ‘the Islamist terrorists’, although there are other groups so it would not differentiate them from the others. ‘Islamist State’, perhaps? It would be accurate, and annoy them by not quite being what they call themselves.
The call for “Why won’t Moslems protest against Moslem terrorism” could be responded to with this:
“We say they are not Moslems, they are Islamists, and we believe what they do is against Islam, not for it” and possibly add “and that is why we say they are not Moslems.“
Does that help? It has helped me.