In Setting→Output→Streaming, you define which track is going out to the stream. In simple output mode it's invisible and hardcoded to track 1 In advanced output mode, you can choose an arbitrary track.
In Setting→Output→Recording, you define which track is going out to the recording. So choose here a track or track collection that doesn't include the track defined for streaming. To define which source goes out to which track, see Edit→Advanced Audio Properties. It's no problem to assign the same source to multiple tracks.