New Jersey was so named because the land it encompasses was an award from the King Charles II to Sir George Cartalet for his loyalty during the English Civil War. Sir Goerge was both originally born on the isle of Jersey and served as a governor there, so the land was eventually renamed, in his honor, by the Duke of York as "New Jersey".
So, in the case of New Jersey, it seems to be the latter.
Source: Dictionary of National Biography, 1885-1900, Volume 09
EDIT - Just found my book on Scottish colonialism to confirm: Nova Scotia (Latin for New Scotland) was so named because while originally settled by the French and called a different name, it was eventually taken over by Scotland, which was technically a separate kingdom at the time of settlement, and renamed.
Source: The Scottish Empire, Michael Fry
In the case of New South Wales, it was named because the landscape was judged to be similar to South Wales.
Since Cook didn't go very far ashore, he had no idea that it was, in reality, very very different.
Sweden had a colony in the Americas in what is now Delaware that they named New Sweden. The area does not resemble Sweden at all; it's flat, humid, and much sunnier and warmer than Sweden.
There were many people from Lincolnshire involved in exploring Australia. Matthew Flinders and George Bass were from there and they were involved in mapping the coast of the continent Official Australian Government site. The land of their home area is very different but they named places after familiar towns and villages. Near Adelaide there are Lincoln, Boston, Tumby, Sibsey, Spilsby, Donington etc
The latter. New York, for example, was founded as New Amsterdam by Dutch colonists, and renamed when it was captured by the British.
In response to this question I see a lot of interesting stories relating individuals or teams of "explorers" to their homelands, but not going any further into the territoriality of these figures. I want to reference back to a section from Benedict Anderson's Imagined Communities, which explores, in part, how conceptions of time in history have changed alongside how we identify with others, specifically at a "national level", and directly relevant to the question:
It was not that, in general, the naming of political or religious sites as "new" was in itself so new. In Southeast Asia, for example one finds towns of reasonable antiquity whose names also include a term for novelty: Chiangmai (New City), Kota Bahru (New Town), Pekenbaru (New Market). But in these names 'new' invariably has the meaning of 'successor' to, or 'inheritor' of, something vanished. 'New' and 'old' are aligned diachronically, and the former appears always to invoke an ambiguous blessing from the dead. What is startling in the American namings of the sixteenth to eighteenth centuries is that 'new' and 'old' were understood synchronically, coexisting within homogenous, empty time. [...] This new synchronic novelty could arise historically only when substantial groups of people were in a position to think of themselves as living lives parallel to those of other substantial groups of people- if never meeting, yet certainly proceeding along the same trajectory.
The parallelism between groups with similar cultural cohesion, especially ones (like in the Americas) with such a rapidly growing (emigrant) population, brought forth a special set of political consequences which included but was not limited to a local "political ascendancy", changes in cartography and navigation, and shared languages of state (aided by print-capitalism). These communities were uniquely seen as privileged continuations of already existing, distant empires, and were relied upon for the benefit of the imperial state.
Sources: Anderson, Imagined Communities, Storey, Territories
TL;DR: "New" toponyms of pre-existing places arose with a changing consciousness of simultaneity and shared history, arguably unique to European colonialism.