Why Do Some Countries Have Presidents While Others Have Prime Ministers?

Imagine two different kinds of schools, one has a headmaster who makes all the big decisions alone, and another is run by a group of teachers who work together. In some countries, like the United States, people elect a president to be their top leader, almost like that headmaster. In other countries, such as the United Kingdom, there's a prime minister, who leads a team called a government, more like those teachers working together.