Maybe. Maybe those meaning were created from its use as the capital…
Well anyway, Australia has a place called the Australian Capital Territory.
For the main part, apart from topology, ACT is Canberra ( a part of the ACT exists at Jervis Bay, separately to Canberra… however this is only as significant as saying the fed gov owns land over here and there , and there. .as they do … )
Well despite the constitution saying that the ACT is for the capital, it seems we stick to the convention of using the urban population name as the name of the capital
First of all, my previous post was not clear on the timeline. According to Japanese Wikipedia, 京師 was used in Chinese as a general term for “capital” and thus to refer to several places in Eastern Asia over time. It was still used to refer to Kyoto as late as the 19th century. For instance, in this famous print by Hiroshige. Wikipedia claims that during the Western Jin era (265–316 CE) 京師 was replaced by 京都 to avoid using the character 師, which had become an honorific given to people, specifically Emperor Sima Yang. This all happened a few hundred years before Heian became the capital of Japan.
This particular section of the article gets quoted verbatim in a lot of places, and I wasn’t really sure where it originates from, but I managed to find a lengthy quote from an essay by a Meiji era scholar that says the same thing.
Hence, it appears that 京都 was used in Chinese as a generic term for capital since the 3rd century, before it became specifically associated with Kyoto. It is used several times in the Records of the Three Kingdoms. It is used also at least five times in the Nihon Shoki, which was written a few decades before the Japanese capital was moved to Heian.
For references on the relevant characters:
師: A gathering of soldiers -> a place where many people gather -> a person who leads many people -> a teacher -> a specialist
By the way, that last link gives a rather NSFW etymology for 京.
It goes on to talk about how even though the character is essentially a drawing of a man ejaculating, it was never used in this meaning. Rather, it started out meaning “venerable.”
Doesn’t look like anyone mentioned India. New Delhi is the capital, but is located in the National Capital Territory of Delhi. The NCT is not just New Delhi, though (it’s made up of ND and 11 other towns and cities that are essentially one large conurbation), so I’m not sure if it counts.