They're the same thing. About the only difference is that "garbanzo bean" is less common in British English than in American English, and chickpea is more common either way, but it's just a language thing.
There are indeed different varieties, with some smaller and some larger, but they don't have common language names like that.
Wikipedia mentions three varieties:
Desi, which has small, darker seeds and a rough coat, cultivated mostly in the India and much of the Indian Subcontinent, as well as Ethiopia, Mexico, and Iran.
Bombay (Bambai), which is also dark in colour but slightly larger in size than the Desi variety. They too are popular in the Indian Subcontinent.
Kabuli, associated with Kabul in Afghanistan. These are lighter coloured, with larger seeds and a smoother coat, mainly grown in Southern Europe, Northern Africa, South America and Indian Subcontinent, having been introduced during the 18th century to India.
Note that it says that one of the smaller varieties is commonly grown in Mexico, so the blog isn't really even quite correct to say "large Mexican chickpeas" (though I've certainly also seen large chickpeas from Mexican brands). I wouldn't lend too much authority to that blog post on that particular point. (A lot of the rest is subjective too, of course.)