I believe that it is somewhat clear that the Jewish ha-Satan, the narrative inception of the "devil", was not envisioned as a red-skinned man with a goatee or Dali moustache, horns, a fur- or barb-tipped tail, a hoof or two, and a pitchfork, or, at the smallest concession to my preconceived notion regarding the matter - if he was, then I have no knowledge of any record of that image being passed into the formative thoughts of Christianity.
Yet, here - or rather, today - he is. Red man, goatee, moustache, tail, horns, hooves, pitchfork. Sometimes slender, sometimes paunchy. So, where did those attributes come from? I'm assuming that they were accumulated over time, rather than all at once - some, like the facial hair and the prominent widow's peak, seem to be rather recent, humoristic or caricaturistic, acquisitions (but I'll concede to correction), while I have no estimation on when the horns and hooves could have been added, except that it must have happened between the birth of Christ, and, say, 1800. So, what are the first sources for those attributes? And why were they assigned to Satan? And by whom? Church officials? Or the general population, who needed someting tangible to fear?
OP is suggesting a potential link to the early formation of Christianity, but might there perhaps also be some older links to the Cult of Dionysus and his companion Silenus?