The head lamps have to be on for the fog lights to work. And, the head lamps have to be on Low Beam for the fog lights to work. High Beam kills the fog lamps. The fog lamps also work when the Parking Lights are on, although this is illegal in most places. Fog lights are "auxiliary" lighting, so they are not allowed to be the primary (only) lights on. Some jurisdictions allow this operation, most do not.

You are right, it is not a simple button, although I am not sure where your troubles are. On my '05, I can set the headlamp switch to automatic, and if I switch the fogs on then the headlamps will be forced on. The beauty of this is that when the key is switched off, the lights go off automatically, and revert to the default operation on the next engine start cycle. I can set the headlights to auto, and if I want to force the lights on, then I can set the fogs on. When I park the car, the lights go off then come back to the normal automatic operation on the next engine start. I never have to turn the lights off. I like this feature.

The fog light switch has to operate a relay that is subsequently turned off when the ignition goes off else the fogs would come on again on the next engine start cycle. You report that you can press the button and the lights come on, but they go off when the button is released. This means that the latching circuit to hold the fogs on while you are no longer holding the button is not working. This is probably the fault of a relay.