All azaleas are in the Rhododendron genus. I have been curious as to what makes an azalea as well. It seems a bit vague and complicated for the general gardener. According to the Azalea Society of America, "In very general terms, azalea flowers usually have 5 stamens, while rhododendrons usually have 10 or more stamens. Azalea leaves have hairs parallel to the leaf surface, usually along the midrib on the underside of the leaf, and tend to be thinner, softer and more pointed than rhododendron leaves. Azaleas flower along the sides of the stems as well as at their ends, while rhododendrons usually flower only at the ends." So by that definition, your plant would not be an azalea.
The fact that you can see more than 5 stamens in the flowers pictured, supports the idea that this is a rhodo, doesn't it?
Wendy, yes the 10 stamens are the reason It would not be an azalea. It's not a good definition though as some azaleas do have 10 stamens. You have to understand them at the level of subgenus to determine which are actually azaleas. It's safest to call them all rhodos unless you have some expertise.