Shortly after Pat Morita died, I read nice bio on him... damned if I can remember where I found it, though.
In it they discuss Morita's inspiration for the character of Mr. Miyagi. Morita, being a survivor of the WWII Japanese American Interment camps (very dark page in our nation's history, btw) based much of his character on men that he knew growing up, both in and out of the camps.
The script intentionally gave a lot of leeway to Morita for how he played the character, and I believe that all here would agree, regardless of the quality of any of the movies (3, 4 specifically), that Pat Morita gave us a very memorable personality to remember.
Another piece of information that you might find interesting regarding the original post. In Karate Kid II, when Daniel-san is looking in at Chozen (Sato's Nephew) you notice a Goju Fist on his uniform (same with later in the film). Since Sato was his instructor and Sato and Miyagi had the same instructor, some conclusions can be drawn.