Should I connect my case fan to a pwr_fan or chassis_fan header?

Well, all three options are possible.

The chassis fan option is definitely the best option - it will do speed control, and will keep the fans running at the required speed. Where possible, this is the best option. Use the chassis fan controllers first

The molex/fan connector connected to a molex connector will have the fan running at full speed all the time, unless you have a fan controller. You'll notice the 'converter' you have only has two pins, since computer fans gracefully degrade down with lesser wires - 2 wires are fixed speed, 3 wires have PWM for speed control and 4 wires also do speed feedback. This will as such, run your speed controllable fan at a single speed, controllable by voltage

As for pwr_fan - various sources source 1 source 2 say that it won't control the fan speed and Its proper use is for monitoring PSU fans where supported. If you have more fans than chassis fan connectors, you could presumably use this as it should be possible to power a fan off it, however. Practically this would be preferable if you didn't want to use a molex connection specifically for fan power, but its not a replacement for a proper chassis fan supply.