I'm aware it's anachronistic but we're playing a game here. That's why it's only for the map and not an actual title. That way the only true empires are BYZ and HRE (possibly others) while the rest are "empires" belonging to a dynasty. In fact, I said that they would still be kings in title but would rule an "empire." Besides, unless I'm mistaken centralisation has nothing to do with being defined as an empire.
Then, what does?
Usually, we call "empire" a political entity that builds a somehow wide state, considering its initial lenght, especially after taking territories out of their "rightful" property by cultural or traditional means.
But it makes no sense that this "empire" tier may bring any administration improvement or bonus of any kind. That's what I don't like about the whole idea of the Fourth Tier.
I thought most people just called states empires when they had extensive territory or interests, at least nominally, regardless of whether or not they had particularly advanced or effective administration. Especially when the territory is not that "traditionally" controlled by the central state, eg., the Chinese and Russians are called empires by virtue of being big and/or populous, but the British Empire only came into being when the British (well, the English) started colonizing and conquering overseas territories which were not part of the British Isles.
Not at that time. Modern historians like the word "empire" and use it for fast-growing warmonger countries, even if they were tribal confederations (the so-called Mongol Empire, which is usually called Khanate, instead of Empire).
Also, you're not considering that in Chinese, the words referring to forms of government might be completely different in their meaning and significancy. For example, "Empire" is a word that comes from Latin
Imperium, which means, more or less, "power over life and death". When the Romans spoke of the lands under the dominion of Rome, they spoke of
Imperium Romanum, more or less meaning "Roman Power/Dominion".
In this sense, every nation has an area under its
Imperium.
Now, the Greek word for the Roman Empire was
Basilea, with totally different ethymology and meaning. Because a language is defined by the evolution of the peoples speaking it, and since English or other Western European languages never got in constant contact with China until less than two centuries ago, there is no actual word in English that could translate the full meaning of the Chinese
Huangdi. There can be a rough translation by the word Emperor, but each language branch has its own titles with its own meaning.
The Russian
knyaz is sometimes translated as Grand Duke and sometimes as Grand Prince. They are kings by all means, in their power and capacities, but traditionally, since the royal title was something important and given by religious authorities at some point, there has only been one Russian king before the Tsar: the King of Galitzia.
It's all a matter of translation and actual meaning of the word, so putting other culture's "Empires" as examples is not a good idea.
In Medieval Western Europe, which is what we're talking about, an Empire was just a descendant of the Roman Empire. There had to be claim. And, if I'm not mistaken, the only serious Empire that didn't claim ascendancy to the Roman Empire was the French Empire under Napoleon. The rest had their "antecessors" reaching the Romans.