I have one of those-works fine. A spinner is used to shape the rivet heads when you are putting the chain together. It is a nice tool to have-but you CAN spread the rivets with a ballpeen hammer or a hammmer and punch.
If you want good results on the cheap get a Granberg "Break-n-Mend" Looks kinda like a pair of vise grips. Unless you are buying chain in bulk though you don't need these tools.
I have a round steel plate that is about 18"dia.x 1" thick on top of my work bench. It is easy and quick to break a chain with that anvil and punch that you pictured if you have a solid bench to work on. Like Stumper said a ball peen will do the same job as a spinner.