Where does the Departure End of Runway (DER) exactly begin?
The picture you linked is from an FAA publication. As pictured the threshold is the end of the runway pavement, it is a thin line that separates usable runway from either taxiway, blast pad, grass, or an emergency overrun area. It is marked at night with a single row of green or red lights (depending on your direction). It can be displaced or reloacated from the original pavment but in this case it will have noticeable markings that include a well defined line marking the exact end. Some ends are usable for takeoff but not landing or vice versa but that is down to individual runways. 

The paint bars near the threshold of an instrument runway are simply for visual reference to help indicate the width and length of landing zone during approach, they have nothing to do with takeoff or departure. VFR runways only have a number and dashed center line, no white bars at all.

To summarize: the threshold is a line not an area, it separates usable runway from non-runway, it may be marked with paint or lights but it can also simply be a change in surface such as tall grass to short grass. Standard departure obstacle clearance climb gradients are calculated with the plane at 35feet above the ground, directly over the threshold at the departure end of the runway.