I totally agree with Danny, and am 180º away from Lane's position.
Fog lights come on with low beam and cast a light pattern horizontally with a cut-off line no higher than 36" at 50' from the car to prevent blinding of oncoming drivers. Technically, then can be left on when switching to high beam, but most car makers switch them off when high beams are on.
Driving lights are only on with high beam and cast a beam of light as a spot (lots of variations, here) 10' - 15' wide at a quarter mile (yes, there's a federal spec for this). Lucas once had a driving/spot light that I loved the name of; Their 8" "Flame Thrower"! I looked like a huge version of a Marshall driving light.
If you have a light that goes off when you switch to high beam, it is a fog or running light. If it comes on with high beam and is not your standard headlight, then it is a so-called "driving light".
Bear in mind that either of these sold on most dealership cars in the USA pretty much suck at doing their job unless the dealer adds a European version, which is much better. In Europe, Fog lights are mounted below the bumper to sweep the road immediately ahead of you (out to 75 feet or so) and use their low angle to enhance road shadows. They also don't reflect much light from fog and snow coming at you, especially if they have amber lenses, because they are aimed horizontally or down. Many in dealership cars can be left on when the headlights are off to further reduce fog/snow glare and still let you see some of what's ahead.
Driving lights are mounted above the bumper to light the road 1/2 mile ahead so you can see and avoid stuff at speed. Since they are always clear, they reflect fog and snow like crazy (but you can't see 1/2 mile in fog or snow anyway, so go back to low beam and slow down!) The only exception to this was the old Perlux lights with the horizontal vanes across the lenses to enforce a light pattern cut-off line as mentioned above.