There is an element of truth in the "Independence" argument but as with most histories it falls somewhere in the middle.
During the treaty negotiations of Ghent the British were aware of the failure of the Lake Champlain campaign and not being able to take Baltimore At that point the realization hit that the War could still be won but a quick knockout was no longer possible. At that point they did not know of anything they could do that would make the Americans sue for an immediate peace.
See this article on the Duke of Wellington for further details
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The British wanted the War over and despite many American claims to the contrary there is no evidence that Britain wanted to reimpose control over the American Republic. The most they were looking for was an adjustment to the borders and if possible a buffer Indian state.
At the same time that Ghent was going on the British were becoming more involved in European Power politics at the Council of Vienna Those negotiations were not going smoothly from a British point of view The British wanted a Balance of Power and a secure Kingdom of Hannover The Prussians and Russians wanted a lot more then Britain would agree to and while it was unlikely, a possible war between the parties was a possibility.
The British taxpayers were becoming more and more dissatisfied with the war expenditures and as the Napoleonic Wars wound up they wanted a return to normality This would include a restoration of the strong trans Atlantic trade.
A continuation of the War against America would cost even more money and men and basically for little return.
So the negotiators were to offer the Americans a generous peace that would restore things to the normality that existed before 1812 Thus the "status quo ante bellum" treaty . As long as they could guarantee "Canadian" territory and restore trade a treaty like that would be acceptable
The real losers would be the Indian allies of Britain who did not have the pull to get their promised buffer state The Indian had to settle for a promise that the Americans would respect their land rights .(… and we now know how that ended) Not unexpectedly the Americans took the offer
The myth of the Second War of Independence started with that offer Monroe was quoted as saying " The United States have acquired a certain rank amongst nations, which is due to their population and political importance….and they do not stand in the same situation as at former periods"
In essence an understanding had been reached between Britain and America to the effect that while Britain could not win a war outright against America they could still drastically destroy American trade and coastal regions Neither side would win in a war like that.
From 1815 on the Britain and America never fought a war against each other and future problems were resolved by negotiations. Both sides spent a lot in preparing fortifications to defend their territory but never aggressively initiated a war .
Individual Americans would try to provoke wars against Canada in the years to come notably in 1837 and the 1860s (Fenian s) but were not supported by the American government.
America had gained its own guarantee of territorial integrity and was accepted by Britain and other foreign powers as a real country That was not the situation in 1783 at the Treaty of Paris This was the most important factor that America got out of the War of 1812,
For further details see The Weight of Vengence: The United States The British Empire and the War of 1812 by Troy Bickham
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