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: