The first stream you will setup as normal with the overlays enabled, whether via the CamOverlay App or the Axis UI.
On the second stream that you are setting up where you wish to disable the overlays, the following should be done:
On the Camstreamer Stream Manager
Audio and video settings > video options > Vapix RSTP parameters and add the text overlays=off
The overlays=off parameter, is usable from Axis firmware 22.214.171.124 whereas the other overlay parameters are only usable from Axis 10.5.2 firmware.
Additional Vapix RTSP parameters can be found here: