I know that the passenger side air bag is supposed to be off if there is a weight of less than 65 lbs. in the seat.
But what is the meaning of the air bag off indicator? In my 2006 MDX it comes on briefly when I start the car and then it turns off. And it does that whether the front passenger side seat is empty or if an adult is sitting in the seat. What good is that? Is something wrong?

 

Just a light to tell you that the passenger airbag is off. When starting the car its just a system check. If it comes on with an adult after starting the car , the system has to be checked for a malfunction.

 

So how do I know if the airbag is on or off for a lightweight 12 year old?

 

The light in your instrument cluster (the guages area) is the one that comes on for a few seconds when you start the car as a system check is done. If it stays on after a few seconds then the system is malfunctioning and you should get it checked.

To your question about passengers, if the system determines that the passenger is too small / light and the system should be disabled then the passenger airbag light will come on. This light is to the right of the navigation screen and different than the one in your instrument cluster.