"Specific" was, at least originally, a generic term to indicate that the parameter being discussed has been normalized by some relevant unit to turn it from an extensive property to an intensive property. Occasionally in the context of specific heat, you will actually see people write out "mass specific heat" or "volumetric specific heat" or "molar specific heat".

People working in a particular context almost certainly just use the term specific heat to refer to whichever specific intensive property is most often relevant, so it doesn't surprise me to hear that some people use it to mean molar specific heat rather than mass specific heat.