It was called, specifically, the war of 1812 insofar as British operations are concerned in the United States. These included the burning of the White House. It was an attempt to retake the US.

The overall war, it’s the Napoleonic war. The US actually tried to stay neutral, though some of her sailors were impressed into the British Navy. This is one of the causes of the war. Incidentally, my fifth grade social sciences textbook in Mexico made the distinction between the European front and this North American front

When in college several of my instructors referred to the Napoleonic wars as the first global conflict. And they have a point.

Britannica seems to have a decent write up of the specific North America. Front

https://www.britannica.com/event/War-of-1812

Also the Paris peace treaty precedes the Napoleonic wars by about a generation. That was in 1783 and it came after several serious defeats of the British expeditionary force, including the last one at Yorktown.