Your pics illustrate the process well.
A different option that a mechanic friend told me about is to skip the screw extractor altogether. Buy LEFT hand drill bits--I found a set of 5 at the local auto parts store. Use a punch so that your drill won't wander when you start to drill (good advice whether you use the screw extractor method or not). If you're using the left hand drill bit be sure to run the drill in reverse. Normally, once the drill gets in a little ways the screw remnant will simply back itself right out. 9 out of 10 times I've used them it works just as described. The other 1 out of 10 I have to follow up with the screw extractor.
The only sound worse than a bolt breaking off is breaking off a screw extractor trying to get one out. They are a necessary tool, but I prefer to avoid having to use them if possible.