Indeed you're right. The pigeon family encompasses many species, all over the world. So much for that argument, eh?
A guy can see photos of lots of different pigeon species at the link below:
http://www.camacdonald.com/birding/Sampler3-PigeonsDoves.htm
And here are a list of pigeon species, an the genera they are part of (there is more than one genus):
http://www.wildlifelands.com/wxs/Taxon4.htm#60
In all over 200 species.