Chris

Being directly elected doesn't make someone a president. Boris Johnson is directly elected, that just makes him the directly elected holder of the office he was elected to ... in this case Mayor of London.

A president has the top job - head of state and in some cases also head of government. Our monarch has the job of head of state and our prime minister is head of government. He/she would still be the prime minister of the government if he were directly elected by us rather than by current means.

However, there are certain complexities in directly electing the PM. The convention is that the Queen appoints as her PM, the one who is most likely to command a majority in Parliament. Turning that on its head would require a great deal of careful planning and law-making, and all we would end up with is a system where the Prime Minister is more likely than at present not to have a commons majority, and therefore less able to pass legislation and more likely to have the annual pantomime of struggling to get a budget approved.