According to this article, which was apparently written in October 2003, it was both. Also notice that they refer to Vernita as "Cobra" (an earlier draft), and mistakenly call Elle "Sidewalker" (though they had it right before in the article). Anyway, there's your answer.
Locations in Japan were primarily used for filming driving sequences with The Bride and a character named Sofie Fatale, played by Julie Dreyfus. The main locations were a desert setting abutting Los Angeles where The Bride exacts her revenge on The Sidewalker, and in a home in Pasadena, California, which served as The Cobra’s home. There were also scenes filmed on sets built in Los Angeles, including the final confrontation between The Bride and Bill. Locations in Mexico provided exteriors for Bill’s hacienda and also for driving sequences.
(Emphasis mine.)
Very well. I know what must be done.