The holes the balls are pressed into have a chamfer on both sides, so drilling a shallow hole just smaller than the shank diameter helps give that material that's been swaged into the chamfer somewhere to go and makes it easier to push the shank out. You can push them out either way, either grinding the back side flush or cut the ball off and grind that side flush. The drill your hole and use your press or BFH. I used a ball joint press that I have with a bolt smaller than the hole I drilled as a pushing tool. The BFH method can get a bit hairy if your impact is not dead on. Mine was about the same as yours, looking like a football if you look at the end of the ball.
I pressed the new ball in place, then heated the opposite end and peened it back into the chamfer with a big ball peen hammer. Or you can run a weld bead around it. There are also bolt-in replacement balls, with thread and nut on the back side.
