What is this valve and how do I fill it? This is the valve in one of the tubes on my bike: 



What kind of valve is this, and how do I fill it?

I originally thought it was a Presta valve, but it doesn't look or work like any of the presta valves I see in instruction videos online. Those presta valves do not have the nut in the middle like on my valve. The nut screws off, but when I do, air immediately starts leaking out and the tube quickly empties. This is in contrast with the presta valves where the inner part can be loosened but still don't leak air unless you press it down with the pump. 

I have a Bontrager Charger floor pump which is supposed to fit both Schrader and Presta, but it does not clamp on to this valve at all while the nut is on, and I cannot fill the valve while the nut is off because in that state the air just leaks back out before I have the chance to screw the nut back on. Here is a picture of the pump head:
This is a Dunlop valve, also known as Woods. These valves always let air in but not out. The nut holds the valve in place, and you are supposed to unscrew it only if you want to deflate the tube.

There are adapters from Dunlop to Schrader, but the most common way to inflate these is a simple hand pump that has just a rubber gasket instead of chuck like shown in your picture.