How can the key be in run and the engine not started?
If the engine were running and died by some other means than using the key to turn it off, (out of gas, pop clutch, etc.) then the engine could be off but the key in RUN.
In normal operation, however, the engine will be running when the key is in the RUN position. But there is also a START key position and this is different from RUN. Many times things are disconnected in START that are connected when in RUN. This is so the engine starter motor gets all the current. You might notice your radio turning off when starting, for example.
