Ok, so we always used a screw driver for this. And we know that you are only supposed to use a screw driver to drive screws, but we never had this tool!
If you have to remove doors from their hinges frequently, then this is an excellent addition to your tool box. It is simple and straight forward, made of good solid bar stock, this tool does make removing the hinge pins easier than that screwdriver or punch that you have used.
The tool has a forked end that is tapered to slide right up under the pin's head with a tap of the hammer. Another tap underneath and the pin is on its way out.
The price is right at only $12.95 and it comes with a life-time guarantee too. It works pretty handy as a mini-pry bar or wedge too.