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.

"power fan" header is usually found on most mother boards. This fan header does not have any ability to change the speed of a fan connected to it. It supplies the full standard 12V to the fan at all times. You won't find any option in the BIOS or ASRock utility to adjust a fan's speed when connected to it.

This header is useful for powering the pumps of AIO liquid CPU coolers that need the pump running at one speed constantly. If you have any fan that you want to run at full speed at all times, or a fan that has its own speed control built into the fan, this fan header is good for those purposes.

I believe the term power fan came from the way some PC power supplies were designed ~20 years ago. The power supply's fan had to be connected to the mother board, or could be for some reason. So the name power fan for the power supply's fan made sense, and has become a tradition for mother boards.