If a king is the leader of a kingdom, and emperor of an empire, why weren't the British monarchs called as such? For example, why wasn't King George III known as Emperor George III? Shouldn't he have the title of emperor?
Because the United Kingdom was a kingdom that made, and ruled an empire. They used "empire" as a loose term to describe all the territories that the kingdom controlled. But the kings never really directly rule over those territories.
They did however add on titles to being king such as "Emperor of India" during the time of the British Raj.