A contour integral can be over any contour not necessarily a closed path (which starts and ends at the same spot. So all integrals over closed paths are contour integrals (as are all normal real line integrals whose path is just some portion of the real line), but not all contour integrals are integrals over closed paths. The symbol the OP asked about is usually just for closed paths.
BTW a double integral with a superimposed circle (or often oval) is an integral over a closed surface. e.g., an integral of a function over an entire sphere.