 ::OED, the most reliable source possible for an English word, has Exonian, first recorded in 1871.  It has no record of Extonian, the nearest related word being extonious (obsolete, rare), meaning astonishing.  As an Exonian of 32 years' standing I have never heard Extonian used.  OED derives Exonian from the late Latin Exonia for Exeter (the Romans of course called it Isca Dumnorum), which appears also in the signature of the Bishop of Exeter and in the designation of University of Exeter graduates as Exon. (Exoniensis)   