> The head of state is the leader of the country. They deal with other heads of states, and typically are in control of the military.
Not quite true, at least not in limited constitutional monarchies, like the UK and Canada (I'm Canadian): The Queen is the Head of State for both the UK and Canada (and for many other countries) and her role is almost exclusively ceremonial (without getting into the role of the Governor General, who represents Her Majesty when she is out of the country).
In both the UK and Canada, the head of government is the Prime Minister, that is, the leader of the party that controls the House of Parliament. In both countries, they do far more than "bureaucratty stuff" since, among other things, they choose and chair the cabinet, set the overall government agenda, etc. The military reports to cabinet through the appropriate minister(s).
(Mitchell Sharp once wrote a book on Canadian government in which he rightly asserted that ours is government-by-cabinet, more than government by parliament....)
In both countries, the PM is effectively head of the executive branch, but definitely not head of state, as well as sitting atop the legislative branch.