Driving lights are different from fog lights, and both are different from "auxiliary" (or "off-road") lights. Fog lights legally canNOT operate with the high beams. Driving lights by law MUST operate ONLY with the high beams. And off-road lights aren't supposed to be used on the streets at all -- my state and several others require them to be covered when driving on public roads, but it's rarely enforced.
on my DD 98 elantra my fogs stay on with the high beam. and their factory. i like the look of them so i never turn them off lol