All the american forces were under him.. Hence making him the Chief Commander....
President is the Commander in Chief due to Washington's reign as the Commander in chief.
Your making a big deal out of the titles. Washington was THE COMMANDER.. all the generals and etc were under him. Thus, making him the Commander -in - Chief.
Here's the definition of "commander in chief:" 1.A head of state or officer in supreme command of a country's armed forces.
2.An officer in charge of a major subdivision of a country's armed forces, or of its forces in a particular area.
Again, please consult a dictionary. You would be right if it were a proper noun in the song, but it is not a formal title, so thus I am right in such context.