Here's a bit from their wikipedia page explaining the short lived name change:
Twice in the 1950s (the McCarthy era), the Reds, fearing that their traditional club nickname would associate them with the threat of Communism, officially changed the name of the team to the "Cincinnati Redlegs". From 1956 to 1960, the club's logo was altered to remove the term "REDS" from the inside of the "wishbone C" symbol. The "REDS" reappeared on the 1961 uniforms, but the point of the C was removed, leaving a smooth, non-wishbone curve. The traditional home-uniform logo was restored in 1967.
I found two copies of this record in a dirty milk crate at a yard sale in Kentucky last summer. One of them was cracked, this is the other one.