The correct answer to this open question is the following.
The role that is exemplified by the President greeting the Prime Minister of Great Britain is the role of Chief Diplomat.
The President of the United States is the most important figure of the US government. He has many roles such as the mentioned above because he is the head of the state and has to maintain good relationships with other Presidents and Prime Ministers of other countries.
The President of the United States is also the Commander in Chief of the Armed Forces, Chief legislator that had the power of veto legislation passed by Congress, and Chief of the Executive branch and its cabinet.