Cooling of the body largely happens via evaporation of water, whether it's the water in your sweat or an external source. There's an additional cooling that happens when doused with water in that some of the heat is likely to be transferred to the water, which then drips away. On the other hand, you're increasing the amount of weight you're carrying, particularly if you've got absorbent clothing on, and you're increasing the risk of chafing from the wet clothes.
On a side note, and partially addressing your edit to include commentary of where you start getting negative effects, sweating works via evaporation. Contrary to the belief of many, being soaked in sweat is not a good thing, because it means that your body can't keep up with the cooling. Ideally, the sweat is evaporating as fast as it comes off of your skin, so you wind up with a faint sheen at worst. Wearing clothing complicates the process because the cloth can trap the moisture and, when you have enough water in the clothing, it acts as insulation, trapping heat in the water on the inner layers. Your body will react to this perceived extra heat by sweating more. If your clothes are already saturated, the water will just drip away, providing some small benefit, but preventing the natural cooling through evaporation. In colder weather, as per your second link, it will also mean loss of proper thermoregulation, and thus an increase of risk of hypothermia, because your body will stop sweating, but the cooling process will continue to occur as water is evaporated off of the surface.
Ultimately, soaking yourself in water when clothed is a bad idea. Soaking yourself in water where you don't have clothes (c.f. topless runners) might have some minor benefit, but will largely be wasted water. If you're training to run completely soaked (say, someone training for a Tough Mudder-type event, or a soldier training for operations in bad weather), then you get the minor effect of learning to live with the side effects, but that's like discussing the benefits of running with bruised feet.
As a postscript, there's a voice in the back of my head saying that pouring water over yourself might also mess with protective oil in your skin, increasing the risk of sunburn and chapping in much the same way that swimming or licking your lips can increase the risk. I don't think that would apply here, since it's a relatively small amount of water that's going down the surface (versus prolonged immersion as in swimming or the pressure of licking one's lips), but it's something to consider.