In the time of empire, British/English were pretty much interchangeable, these days we make a distinction.
The President's immediate correction was spot on, and all the sources critisizing it (daily mail etc) are formally wrong.
There are no British Embassies, because there is no country called Britain or Great Britain. Great Britain is a geographical entity, not a political one. Strictly,
British refers to England, Ireland, Northern Ireland, Scotland, Wales and the other British Isles.
The embassies we have worldwide are called
Embassy to the United Kingdom, [insert name of city], just as the President said.

edit:
But Unlike the president, I remember which country is my country's best friend
you mean France right? you know, that country that helped you win independence from the English. dont worry I'm not offended
