In 1934-5 Slingerland came out with the Broadcaster. Then, oops, Gretsch said that’s our brand name, so Slingerland dropped that model name and the Radio King snare drum was born c.1935-6. Thanks to Mark Whitey Cooper for the clarification.
Notice that the Broadcaster has a Todd 1 tone control. (thanks Brooks Tegler). The RK apparently has a Todd 2 tone control. Notice the tab springs under the ends of the snare wires on the Broadcaster. This pushes them away when the strainer is disengaged.
Of the 2 badges: The darker one is the Broadcaster.