Sometimes it takes more than gentle force to pop off an old lockring.
This is the tool of choice used by old timers who do lots of these. If you don't want to buy or borrow one, consider letting the LBS do this job, or simply set an old dull chisel into one of the notches and giving it a sharp hammer blow. Don't tap it, that just drives the chisel in, give it a meaningful shot which should sent it spinning off.
Once the lock ring is off you can try the cup, but if it's rusted in there's a good chance that it'll shear the pins off a two pin tool. There were 6-pin tools back in the day for stubborn cups, but i don't know if they can be had today. If 2-pins isn't enough, and you have a small home shop, you can make a 6-pin tool yourself from scrap metal, or resort to desperate measures. Try driving the cup out with a chisel or punch set into a hole, or try grabbing the cup with a big pair of Channellocks, or try clamping the cup in a vise and turning the frame.
If it's really seized, you might try getting the right side off first (Don't forget that many are left hand threaded) then using a fine thread 9/16" bole and nut to work the left off.