As far as I know, only 3.5 presented Athas with the "equivalent" of copper pieces in the form of lead beads. In no other edition have I noticed such a, weird IMO, currency.
The 2nd edition setting, as well as Dragon's 3e take abbreviate ceramic pieces as cp (4e did away with the original currency). While new players might get confused of this with copper pieces, I believe any Dark Sun fan would tell the difference. Additionally, copper is not used as currency (you have gold, silver, ceramic, and bits). What you've probably seen is 3.5e and Gabriel's 5e iteration, where Cp is used, instead.
Longswords and shortswords in PHB, and subsequently in my Money and Equipment of Athas, are actually generic terms for swords of any type, shape, and weight that fit their range. Historically, the above terms are ambiguous; I haven't included any of the "specialized" swords for the sake of convenience, and in accordance to 5e's simplicity. A macuahuitl is a type of sword tied to a specific culture, in our case the city-state of Draj. You could easily call all the swords that the Draji warriors carry around macuahuitl, but use the stats for longswords. Likewise, the elven tribes craft and use long/shortswords but they would call them elfblades or, in their language, uta.
Bear in mind that I've substituted some PHB with athasian counterparts. Carrikal, for instance, fits the role of a battleaxe perfectly, Datchi club is actually a greatclub, and the dreaded Impaler resembles a war pick.
Thanks for your comment, I'd appreciate more feedback!