mid-15c., "full of pride," from Latin fastidiosus "disdainful, squeamish, exacting," from fastidium "loathing, squeamishness," most likely from *fastu-taidiom, a compound of fastus "contempt, arrogance" and taedium "aversion, disgust." Early use in English was both in passive and active senses. Meaning "squeamish, over-nice" emerged in English 1610s.
I think the distinction shouldn't be made at all. Fastidiousness is a strong dislike towards disorder - whether through lack of care in finish and arrangement, or through untidiness, dirt, and unseemly features. A fastidious person will be equally appalled at the corner of shirt sticking from under the belt as for a smudge of dirt on their hand. The differentiation between untidy and unclean is insignificant here - the dictionary definition doesn't cover two different traits, it covers two symptoms of the same trait of character.
Note the word has different connotations, starting with positive (very good attention to detail) through neutral (conditions must be fulfilled precisely) to quite negative (impossible to please, spoiled) but they are all still related to the same trait of character, which may be desirable in certain conditions and unbearable in others.