Sleep, standby and dark wake
Sleep, standby and dark wake describe different power states. Mac sleep reduces system activity; display sleep blanks the screen. A monitor's standby state is managed by that device and need not disconnect it from macOS.
Dark wake is a background wake of the Mac while its screen remains off. It can be followed by a full user-visible wake, so it should not be treated as proof that the desktop is ready for ordinary interaction.
BetterDisplay's wake and brightness options may respond differently to these states. Turning off a backlight, dimming an image and logically disconnecting a display also have different effects. See Software XDR/HDR Upscaling and Dimming and Soft disconnection.