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You have the wrong trigger for your second automation.
You are not checking to see if the motion sensor binary_sensor.family_room has been off for 2 minutes, you are checking if light light.family_room_right_level_on_off has been off for two minutes.
Also note that because you have a 160 second transition to off, the light will not turn back on if motion is detected during this time. If that is going to be a problem, shorten the transition time. Transitions can not be interrupted. It’s annoying but that’s the way it is.