"Hellenistic" is a modern word and a 19th-century concept; the idea of a Hellenistic period did not exist in Ancient Greece. Although words related in form or meaning, e.g. Hellenist (Ancient Greek: Ἑλληνιστής, Hellēnistēs), have been attested since ancient times, it was J. G. Droysen in the mid-19th century, who in his classic work Geschichte des Hellenismus, i.e. History of Hellenism, coined the term Hellenistic to refer to and define the period when Greek culture spread in the non-Greek world after Alexander’s conquest. Following Droysen, Hellenistic and related terms, e.g. Hellenism, have been widely used in various contexts; a notable such use is in Culture and Anarchy by Matthew Arnold, where Hellenism is used in contrast with Hebraism.
What initiated the term Hellenism?
Geschichte des Hellenismus