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All barlows are achromatic.
A 3 element is a design that allows the focal length to be reduced (i.e. the tube shortened) without severe curves on the lenses.
Those severe curves could also lead to other aberrations and higher manufacturing costs.
Hence, 3 element barlows.
If the negative lens is followed by a positive lens, the rays can be brought close to parallel, which:
--reduces vignetting
--doesn't expand the eye relief
--keeps the magnification the same at varying distances from the lenses
This type of barlow lens is called telecentric, and it can have better optical qualities than a conventional Barlow.
Examples from TeleVue, Explore Scientific, and harry Siebert, and, in the past, Meade.
But note that the 2-element and 3-element barlows perform roughly the same, but one is shorter than the other.
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