You are obstinate to the point you refuse to believe what your eyes are telling you. The light on the podium goes out as she walks away. If the light on her podium is a reflection, then you'll have to tell us where that light came from.
Actually, it seems to be illuminating her face.
You understand how TV lights work, right?
Go spend twenty years setting them up, taking video in them, and dealing with idiots walking and moving in and out of shots.
Then we can talk.
Otherwise play around with a flashlight and some shiney finishes. This is not rocket science. But it is physics.
It is definitely illuminating her hand!
From another angle, you can see that Trump's desk surface is angled downward from front to back, yet Hillary's is not, it's flat with no drop at all. This allows the greatest viewing angle down into the screen.
I've done stage lighting before. It's no longer advisable to do lighting from below the actors as it gives them a ghoulish, unnatural look. Foot lights are long gone from modern stages for that reason except for special purposes. You especially do not use low lighting on someone with wrinkles on their necks.
I notice other slight irregularities about Hillary's podium that are not present in Trump's podium that have to be there to accommodate the presence of the device. For example the ledge at the front of the podium with the curved front on Trump's podium is perfectly flat, but Hillary's podium has a slight 3º to4º angle upwards toward the rear to clear the upper edge of the screen.
I got curious what was on that screen area on Trump's podium so I enhanced area from the image above. Here's what I was able to bring out and there is certainly nothing there that can reflect anything.
As you can see, what ever is down in there is recessed further in and I think I can make out a clock, perhaps a timer for his speaking. But certainly no screen or teleprompter. . . and no flat reflective surface.