You have to understand that Buddhism blends wherever it goes. Although the core teachings remain intact, but the outer bark--the "form" that the religion/philosophy expresses--adapts to the receiving culture. Therefore, iconic representations of ideas or teachings differ from area to area. Tibetan Buddhism has a unique flavor, and so do Japanese Buddhism, Chinese Buddhism, Thai Buddhism, Sri Lankan Buddhism, and so on.
You can think of Buddhist teachings as a vehicle, a car. You'd have all sorts of models, parts, designs, etc., but a car is still a car despite all its variable forms to choose from.
To get back to your question, the Fat Buddha ("Laughing Buddha") has been an image exclusive to Chinese Buddhism. It has come to represent good luck and wealth, which reconciles amicably with the Chinese merchantship. As for the actual, historical Fat Buddha.. we don't know if he did exist. He's as far back as a legend. I could be wrong though, about the legend part; I'm not an expert in Chinese Buddhism. You have to go dig history.
The Buddha--Siddhartha Gautama--has taught, and lived, a life of moderation so it is insensible for people from any culture to portray him as having a fat, gluttonous belly.