From Latin 'versio' (a turning) — originally 'a translation,' a turning of text between languages.
A particular form of something differing in certain respects from an earlier form or other forms of the same thing; a translation of a text into another language; an account of an event from a particular person's point of view.
From Latin 'versiōnem' (accusative of 'versiō'), meaning a turning, a turning around, a translation, from the past participle stem 'vers-' of 'vertere' (to turn), from PIE *wer- (to turn, to bend, to wind). The original and primary Latin sense of 'versiō' was a translation — the turning of a text from one language into another, as though rotating it to face a new audience. This literal use persisted in English from the 1580s through the 17th century
The phrase 'the King James Version' preserves the word's original meaning: a 'version' was specifically a translation of a text — a turning of the Bible from Hebrew and Greek into English. The broader modern sense ('a version of events,' 'software version 2.0') grew from this translation sense by generalization. Every version is still, etymologically, a 'turning' of something.