Windows user here. I don't see the second window moving its position with Lights Out though I do recall seeing this as a bug in the past.
If second window is set full screen, on secondary display: it behaves as expected with Lights Out.
But if the second window is set not fullscreen, then Lights Out dimming or blanking makes that whole thing temporarily disappear. Wherever it is located, and on whichever display screen.
If the main screen is set not fullscreen, and the second window is set fullscreen and on the same display, hence entirely covering main window - I got curious - weird results from Lights Out. An impractical use case! To extricate from this silly condition Shift+F11 returns the second window to its non-fullscreen mode, and then control can be regained.