From All Things Hair on The Pixie Cut: A history
The flapper cut
Short hair leapt into the mainstream around the ’20s, exuding charm and magnetism (as anyone who has a soft spot for Gatsby-esque sparkles and flapper dresses will know). The party-fuelled era saw the short haircut come into popularity as a symbol of women’s newly found independence. Having long hair meant being shackled to old stereotypes of femininity while the pixie cut (and many other short hairstyles) oozed fun and freedom!
In answer to your question, the length and style of pixie's hair has little to do with the origin of the name of this style. It was simply called the Pixie Cut. Likely since it made the ladies feel daring and free, perhaps elfish. You will look in vain for little forest people sporting this fashion. They have their own fashions they do not share with others.
Again to answer the question "why was the short haircut called pixipixie?" The origin is from the hair style so named. It isThe woman was called pixie because itshe was given a Pixie Cut. There is no more reason than that. Sorry if it seems tautological but it is like a brand name. No justification or source before it exists. No one trapped a pixie and copied its hairstyle.