It's pretty well documented that the Americans wanted greater control over Canadian territory and were generally upset with how the British treated America as a power. What I would like to know what the British were thinking when they were attacked. Was it an entirely defensive war for the British or did they want to take back the colonies?
It was entirely defensive. The British were fighting the French in the largest war on Earth at the time, and didn't want to devote resources to protect Canada. The American declaration of war and subsequent invasions forced them into the war. Given the choice, they'd have stayed out of it.
I go into more depth here.
u/PartyMoses hit the nail on the head. The British war aims pre-1814 were almost entirely defensive. Isaac Brock did seem to embrace the "the best defense is offense" mindset. But the British goal was to protect Canada while not diverting too many resources from the war in Europe.
Post 1814 the British tried to hard to pressure the Americans into creating an Indian buffer state in the Ohio Valley, and to allow British navigation rights in the Mississippi. But both were secondary to the defense of their colonies in what is now eastern Canada.
There is no evidence at all to suggest the British wanted to re-take the American colonies.