I just did the fog light mod on my 2004 V6 Forerunner and my DRL do not stay on, only fog lights and turn signal stay on. I need to mention that I did have to replace relay because I did break off a tab on the install. The relay is not OEM due to cost, I purchased at O'Reilly's.
I'm a little bit confused, it looks like your DRL's (amber lenses toward the left and right sides of the vehicle) are staying on, and your fog lights (clear lens toward the middle of the vehicle) turn off, which is what is supposed to happen.