My EB says

Diatonic: a musical term signifying literally "through the tones", otherwise music in which the notes employed are confined to those of the key, major or minor, in which it is written. Hence diatonic music gives a general impression of strength, simplicity and solidity as distinguished from the more restless and poignant character of chromtic music in which notes from foreign keys are introduced by accidentals. The diatonic was one of the three Greek genera of scale.

The phrase "notes ... confined to those of the key" is what I always remembered from my Music Theory days...

Robin