In the U.S.A. much of the popular conception of the "Buddha" is (mistakenly) the Hotei/Budhai you are talking about.
Much of this comes from the Chinese workers who first introduced the "Buddha" in the late 1800's in California and Hawaii then.
That "Jolly Chinaman" figure as it became known was the first introduction to Buddhisim...Chinese style...that many people in Hawaii or California had.
So they identified that figure with the "Buddha" figure.
Shame on you Shakyamuni for setting the precedent of leaving home.
Did you think it was not there--
in your wife's lovely face
in your baby's laughter?
Did you think you had to go elsewhere (simply) to find it?
from - Judyth Collin
The Layman's Lament
From What Book, 1998, p. 52
Edited by Gary Gach