How can I define the term empire and show how useful the term is through describing major states? Can my argument be that: The term empire is weak because it has broadened and evolved over time (Formal territorial empires then informal empires), and it has also been regarded as a negative word (used when criticizing states and oppressive countries that take over defenseless nations) which makes it an outdated term (no longer used as no modern state regards itself as an empire)
I agree that 'empire' is usually used as a negative term today. No state today calls themselves an empire because the term has so many negative connotations. 'Empire' was not always thought of as negative. Empires were once a source of national pride.
Is the term 'empire' weak? Certainly it means different things to different people. An empire could be territorial i.e. one people ruling other, subject, peoples. Examples would include the British Empire, the Russian Empire, the Roman Empire, etc.
An empire could also be a governmental structure, a form of monarchy in which an emperor has power and authority over kings. Examples would include the German Empire of 1871-1918 in which the king of Prussia established his authority over other German kings like the king of Saxony and the king of Bavaria. The Roman emperors had authority over several subject kings, most famously King Herod of Judea but there were others. The British Empire recognized several subject kings such as the king of the Zulus in South Africa. Morocco continued to have a sultan even after Morocco was incorporated into the French Empire. In this sense, an 'empire' is ruled by a king of kings.
Sometimes 'empire' has been used to attribute greatness to something that really does not deserve it. Someone once said the Holy Roman Empire was neither holy, nor Roman, nor an empire. The HRE was a very loose confederation in which the emperor had very little real authority over his many subject peoples.
Napoleon probably called himself 'Emperor of the French' because so soon after the French Revolution he could not have gotten away with calling himself the king of the France, but to all intents and purposes, that is what he was - the king of France.
'Empire' has also been used figuratively. For instance some captains of industry have been said to rule over financial empires.
So long as 'empire' is used to generally describe something that is big, powerful, overbearing, etc. the term is fine. If one were to apply 'empire' to something more specific, you start running into problems.
Part of your problem is the same problem anyone has who attempts to apply modern standards or morality to the past. Just because people in the past may have been proud to live in an empire, does mean these people were necessarily evil. An imperial structure may have been the best political structure available at that time in history. Much good has come from empires. The British Empire spread democracy around the world and gave much of the world a common language and similar concepts of law. Napoleon gave the world the metric system. The Romans gave Europe a unity that had not previously existed.
Hope this helps, C.M.