if Bill's definition of equinox is correct "all parts of the earth" then it'll be the same all over the world.

making night and day of equal length in all parts of the earth

This is true. At the equinoxes the length of the day equals the length of the night all over the world.

in the Northern Hemisphere the vernal equinox occurs about March 21, the autumnal equinox about September 22

This is also true. in the Northern Hemisphere is a prepositional phrase modifying the rest of the phrase. Therefore what follows is only true for the Northern Hemisphere. The language is called English. I realize that you don't use it in other parts of the world with the purity that it still has among USns but there should be some glimmer of recognition even there where you are. It is, however, true all over the world that in the Northern Hemisphere the vernal equinox occurs about March 21, the autumnal equinox about September 22.