When did Asian come to mean only certain parts of Asia such as the East?

As your quote from Wikipedia indicates, the extension of the term "Asia" to the current geographical expanse that is thought of as one of the continents of the world happened a long time ago. In current usage, the noun "Asia" refers to a region of the world that includes both the "Middle East" and the "Far East".

There isn't any etymological basis for giving a narrower sense to "Asian", "Asiatic", "Orient" or "Oriental". "Asian" and "Asiatic" are just adjectives built on the same base as the continent name "Asia", and "Orient" and "Oriental" are just derived from a Latin word meaning "east(ern)". All of these terms have been used to refer to nations and peoples of the Middle East as well as those of East Asia and South Asia.

The distinctions in usage and connotation that have developed between these terms are based on social phenomena, not on etymology. There isn't any single "correct" definition of Asia, Asian, Asiatic, Orient or Oriental because all of these terms refer, not to natural categories of prexisting things, but to made-up concepts that people have thought differently about in different times and places.