"Run" and "On" are different names for the same key position, where the engine keeps running if it's started and where the idiot lights all illuminate when you first switch it there.
Typically you'd have some permutation or combination of Lock (lock steering column), Off (nothing happening), Accessory (Radio works but engine doesn't), Run or On (Engine works, idiot lights come on when first switched there, some additional accessories are alive), and Start (crank up the motor, often also disconnecting various non-essential accessories). You can have the key in On/Run without starting the engine, and in many cases that would activate the battery combiner solenoid.
If you have a BIRD sort of device, that works by detecting votlage levels, it might not connect the batteries under normal conditions until the engine is actually running and the alternator charging and the chassis voltage higher than some threshhold.
