Ok, now this gets strange. I had "Disable unredirection of windows" off (=grayed, switch towards left). Then I checked it (switch towards right), restarted Firefox and nothing changed - tearing was present.
Closed Firefox, switched back the setting as it has always been (the switch towards left). BOOM, after restarting firefox tearing seems to be gone. Have tested several streams successfully - hope it stays like this. Really cannot understand what just happened. Thanks so much bwat47 for your help!