I essentially do what Ellis recommends, although I usually use a somewhat different method:
Draw two circles
Copy and paste in place
Select the topmost circles
Use Object -> Pathfinder -> Intersect, which creates the green object in Ellis' screenshot
Then grab the bottom-most circles and use Object -> Pathfinder -> Exclude Overlap
Lastly, with your Exclude Overlap circles still selected, use Release Compound Paths.
This has the same effect, but I don't love the anchor-points-and-scissors method myself. Many years ago I taught myself Illustrator with a flaky mouse on an irregular surface; it left me with a pathological hatred of tasks that require fiddly mouse-work. So I have keyboard shortcuts for all of the above, and can select all of the objects without having to very carefully mouse over a path. If you're fast with fiddly mouse-work, Ellis' method will probably work better for you than mine.
I think it would even be safe to stop after the first pathfinder operation when you have the overlap colored. I don't see a need to exclude the overlap on the objects below, but I would group everything so they move together.
You're completely correct, Peter. I set up these shortcuts for a project where I had to make a large number of multicolored Venn diagrams - there were no empty circle-fragments, they all had a transparent tint of some sort.
If you only need the overlap colored and can leave everything else empty, much less work is required.
Even if the two main circles are colored I don't see any need to mess with them. You are making a copy of them for the overlap, so they can remain just as they are, two overlapping shapes, with the overlapping intersection that you just created, all grouped.