I must ask because I can't find an answer, it's a bit of a soft question and for that I apologise.
Why are they called contour integrals? In all the subheadings I've checked there is no special reason.
My "guess" is that a contour cannot cross itself (say you a curve doing an 8 shape, as a contour it's actually two o shapes that touch)
Now I thought contour was the name given to isolines (iso comes from Greek meaning "the same) when used to show terrain.
I'd extend this guess to say "isolines are the closure of a level set" and a "contour is (directed?) subset of an isoline, such that no point is visited twice unless it is the start and end points, which must be the same"
Unless "contour integral" has some implied complex meaning (I know it occurs in complex analysis, but does it have to be a part of? Line integrals have no problems when they cross themselves, so I think a contour is slightly stronger than a "closed curve"?)
Sorry to write so much about something so little! It's just it doesn't seem special enough to be worth its own name (contour integral), so I speculate that contours might be special, and contour integrals are the integrals over these special things.
No, contours in contour integrals are allowed to cross (at least in modern formulations). If they don't, they are called "simple contours". On the other hand, it could be that Cauchy originally formulated these things for simple contours only.