That's not denial of the existence of other Korea, and not "we're the true Korea"; it's more that there is only one Korea, it's the entire peninsula, and this unfortunate state of affairs where there are currently two separate governments isn't an indication that there are two Koreas. There's one Korea. The whole thing.
I asked the guides specifically so I think I've got it more or less right.
In South Korea, they call South Korea "Hanguk" which means "Land of the Han" (Han people being the Korean ethnic group). They call North Korea "Bukhan" which is just "Northern Han."
In North Korea, they call themselves "Joseon" which means "morning calm" and was the name of the last Korean kingdom.(cf. http://en.wikipedia.org/wiki/Joseon) Meanwhile they call South Korea "South Joseon."
So, they each call themselves and the other "Korea" in a sense, but they disagree on what the word for "Korea" is in Korean.
"Korea" is itself a romanization of "Koryo," which was the name of a much earlier Korean kingdom.
For more, see http://en.wikipedia.org/wiki/Han
Similar situation for some years in the Taiwan Strait, where the not-a-republic People's Republic of China faced down the also-not-a-republic Republic of China (Taiwan).
Still, if it's a case of wishful thinking, why "democratic" and not "The Awesome, Glorious Empire of North Korea Death to America"?
So (for a not-quite-correct example) things would progress Congo -> Republic of Congo -> Democratic Republic of Congo -> People's Democratic Republic of Congo.
A "people's democracy" is a country in transition from bourgeois democracy to socialism. In this stage there is more than one class, the largest being the proletariat, led by the Communist Party, which is therefore the dominant power.
The other common country name includes "people's republic." In both cases, it's important to understand the definition of "people". People are communists. Look up "People's dictatorship" for more information on this. Essentially, a "People's republic" is a country that is ruled by a communist party.