Cincinnati were the Red Stockings and Red Legs, yes, but that was in the American Association, and prior to entering the National League in 1890. https://en.wikipedia.org/wiki/Cincinnati_Reds#The_National_League_returns_to_Cincinnati_.281890.E2.80.931911.29
They were often called the Redlegs in the media prior to that, like the Yankees are called "The Bronx Bombers." The team asked that the media either use 'Cincinnati' or the old name 'Redlegs', but the team never officially adopted the nickname. The article even states it was a "preference" as the team didn't want to be associated with communism or the Red Scare of the McCarthy era. It was never official and the fans never picked it up in colloquial use. Is "Jays" official for Toronto? Everyone uses it. Or "D-Backs" for Arizona? Both are more common and popular than "Redlegs" ever was in the 1950s.