To be quite exact, it means "near the capital", namely Kyoto/Nara. Osaka is the economic powerhouse of the region but has never been the capital.
These days the more common name is Kansai (west of the toll gate), to contrast with the Kanto region (east of the toll gate), aka greater Tokyo.
There is a university there whose name would directly translate to "Kinki University". They opted to editorialize it a bit before going international (they go by "Kindai University" outside of Japan).
From the makers of the PIN number and the ATM machine, it’s Kinki U. University.