Islamist - islamicist
An islamicist is a person who studies the religion of Islam (adjective Islamic). A follower of the religion, a believer in Allah, is a Muslim (sometimes spelled 'Moslem' in the past). The word Islamist has been used for 'Muslim', although it seems quite unnecessary to AWE, and it is unrecorded in the OED 2nd edition. In recent years 'Islamist' as an adjective has come to be used almost exclusively with the meaning 'supporting or advocating Islamic fundamentalism', and as a noun to mean 'someone who supports or advocates Islamic fundamentalism'.
Certainly, it is an error to use the word islamicist to mean a believer in Allah.