central city vs city center

central city

noun
  • Synonym of central business district 

  • Used other than figuratively or idiomatically: see central, city. 

  • The main city of a metropolitan area; particularly one containing multiple cities. 

city center

noun
  • The central business and shopping area of a city. 

How often have the words central city and city center occurred in a corpus of books? (source: Google Ngram Viewer )