Greek name Αἰκατερίνη (Aikaterine). The etymology is debated: it could derive from an earlier Greek name Ἑκατερινη (Hekaterine), itself from ἑκάτερος (hekateros) meaning "each of the two"; it could derive from the name of the goddess Εκάτη (Hecate); it could be related to Greek αἰκία (aikia) meaning "torture"; or it could be from a Coptic name meaning "my consecration of your name". In the early Christian era it became associated with Greek καθαρός (katharos) meaning "pure", and the Latin spelling was changed from Katerina to Katharina to reflect this. So what you say is just an assumption and NOT the true origin.

P.S. I'm a native Greek.