I found this. There are many types of papaya. And I was wrong about the Mexican papaya. The ones we know (or I have been calling) as Mexican are actually Hawaiian.
Hawaiian Papaya Information, Recipes and Facts
This is the Mexican one, notice the shape is different:
Mexican Papaya Information, Recipes and Facts
And what I have been calling Thai, is not, it is in fact called the green papaya that actually originated in South America, but most used in Thai and Viet cuisine:
Green Papaya Information, Recipes and Facts
Notice that shape is different still.
Lastly there is another! Strawberry papaya!
Strawberry Papaya (Sunrise) Information, Recipes and Facts