Doc John at World of Psychology enumerates the 9 different flavors of love. He's a little disappointed in the study being discussed, as they only interviewed about 50 people.
My personal favorite, not listed in the big 9, is Hassliebe (literally hate-love), the German expression for tough love toward children.
A quick look through Shakespeare or Pedro Amoldovar would probably yield a few more flavors.
The original story, in the Guardian, is here.