Reus airport is over 50 miles direct distance WSW from Barcelona city centre. . In terms of latitude it is south of Barcelona by only a few miles. Far more west than south. To me "Barcelona South" suggests a location offshore !
That name does sound like a hangover from earlier Ryan Air times. (as per Airport Planner 1 above) The airport refers to itself as plain Reus as far as I am aware. Costa Dorada could well be a better and more accurate marketing name. If SEN are dropping "Barcelona" from the name, then good. At this stage of its development it hardly needs to be accused of the kind of misleading information for which Ryan Air were notorious.