Ok, everything I've seen refers to Manchester Central being outside the football league and the two Manchester clubs inside the league (Ardwick and Newton Heath) not bearing the Manchester moniker so Ardwick deciding to change to Manchester City FC after a reorganisation. Nothing about the two clubs existing simultaneously. In fact in the 1893/94 season they played the first 27 games as Ardwick AFC (who were dissolved after the 27th game) and the final game as Manchester City FC (who were playing their first ever fixture), suggesting that the two clubs didn't, in fact, exist at the same time.