The difference between fog and driving lights has nothing to do with color.

First, the lens color is not an issue at all. Traditionally (i.e., in the older days), fog lights used to have yellow lenses (not driving lights, as you wrote), but newer (yet still decades old now) research showed that the coloring was really useless and actually reduced useful light -- for the source of this info, you can refer to any recent edition of Bosch's "Automotive Handbook".

The actual difference (again, you can refer to Bosch's book as a source) between fog and driving lights, which share the same reflector, is the front lens' optical design as well as a cutoff shield:

Fog lights have more fluting (vertical sections of lenses) that spread the light horizontally for more side-to-side coverage, as well as a shield or hood over the bulb to create a sharp horizontal cutoff on top of the illuminated field (almost the same as the low beam cutoff on Euroheadlights (such as you installed) or e-code (sealed beam substitute) headlights for older cars. Fog lights are strictly for very short distance illumination of the road, and thanks to the horizontal cutoff built into the lamp, can be used in conjunction with low beams without blinding other drivers.

Driving lights have less fluting in the front lens, to leave most of the light directed forward by the reflector continuing in the direction; and there is no shield for a horizontal cutoff. The beam spread is less, more the light being directed farther forward for a longer range illumination of the road. Generally, it should be wired with the high beam, to go off when you switch to low beam.

A third type of light is called the "pencil beam", and has an even more intense illumination of the road farthest ahead by having no fluting at all, leaving all the direction of the light as a function of the same parabolic reflector. It should only be used on racetracks with long straight stretches. So think of driving lights as an intermediate-range illumination.

So, as you can see, fog lights and driving lights are not interchangeable based on just front lens color, nor the front lens' optical design alone; the shield (on fog lights) is a vital, integral component built-in as well.

Hope this helped.
And buy a copy of Bosch's book to further explain these things.