Even from your first post you don't really understand how this war plays into the creation of Canada. It wasn't just a war fought by Great Britian, in fact most of land battles were fought by Canadian militia and Natives( in 1812 and 1813), who were out numbered by their American foe. This war also plays into nationalism whos view of history you want to believe more. For Americans its that you won or that it was draw. For Canada it was win because a small group of armed colonial troops kept you have invading and taking over the Canadian colonies. many American fail to remember that was part of war, to put of North American under the US flag... the US lost. British never had a goal of expanding Upper and Lower Canada, just keeping what had, and in that they succussful.
As a Canadian National, I know that Canada won this war. Americans just have a hard time at looking at the facts and realizing that yes indeed they lost to a handful British Regulars, Canadian Militia, and Canadian natives. Just by being able to get a treaty signed and the war stopped is a victory, because theUS was over 10 times larger than Canada and was unable to take over. The Battle of New Orleans, doesn't factor in this because it was fought after the peace treaty and its effects therefore didn't play into the treaty, as the war over by 1815.
The war ended in a stalemate with, the Treaty of Ghent, signed in Belgium on December 24, 1814, returned all conquered territory, so that the situation was the same as it had been in 1812. The big difference was that the people in the British colonies, especially Upper and Lower Canada, had gained a sense of nationhood. Together, they had fought to repel an invader, and this gave them a new pride and caused them to think of themselves as Canadians, identifying with the land in which they lived rather than in the countries they had come from. THis why from a Canadian view point the war was won by Canada.
Here is quick list of the battles
JUne 18 US declares war on GB including Upper and Lower Canada.
JUly 12 US forces cross Detroit river
July 17 capture of michilimackinac ( US lose)
Aug 16 Siege of Detroit by 600 Can and Native forces, it surrenders with out a fight.
Oct 13 The British, and Can Militia led by Brock and General Roger Sheaffe, repulse an American invasion at Queenston Heights. Brock is killed
Apr27 American fleet on Lake Ontario captures York [Toronto], burning the Parliament Building
May-DEc Americans take Fort George but are driven back at the battles of Stoney Creek (June 6) and Beaver Dams (June 24)
Sept 10 American navy defeats British at Put-In-Bay on Lake Erie
Oct-NOv American advance on Montreal defeated at Châteauguay (Oct 26) by Colonel Charles de Salaberry, and at Crysler's Farm (Nov 11) by Colonel Joseph Morrison
July US forces again invade Niagara peninsula but defeated at Lundy's Lane (July 25)
Aug British troops land near Washington, D.C., and burn the White House and Capitol building
Sept The British, led by Sir George Prevost, strike south, but are defeated at Plattsburg on Lake Champlain
Dec 24Treaty of Ghent ends the war and restores the boundaries to their positions at the beginning of the war