score:55
First of all, as the definition you cited states,
The term empire does not have a precise definition.
The Aztec Empire was large by the standards of their time in their part of the world. It dominated the Valley of Mexico and was a major power in Mesoamerica generally. Land size is not really a indicator of imperial status per se, but in context, the Aztecs was the largest player around.
Also, I think you've severely overestimated how big European duchies were. See for instance: Salzburg, Limburg, Silesia, Bukovina, Bar, Bremen, Holstein or Guelders.
Secondly, Aztec rule consisted of an imperial power (Tenochtitlan, Texcoco, and Tlacopan) subjugating and exacting tributes from foreign cities. This bears fairly close resemblance to classical empires. For example, the Athenian Empire through which Athens ruled the Aegean shores.
In fact, a powerful imperial core exercising indirect rule or suzerainty over autonomous provinces is very typical of imperial political structures. For instance, large parts of the British Empire was controlled through indirect rule. Notably, the princely states of India.
My previous two points addressed the substance of an empire. Ultimately, and perhaps more importantly however, the name of a political entity is primarily a matter of style and form. The best example is the Holy Roman Empire, which is neither holy, nor Roman, nor an empire in practice.
The Aztec polity was headed by the huetlatoani, a title which outranked the tlatoani who ruled other contemporary Mexican city states. By analogy, therefore, the title of huetlatoani is translated as "emperor". Since the Aztec Triple Alliance was ruled by "emperors", it was an Aztec Empire.
Upvote:5
I think it is called an empire by the following criterion: it is a multi-ethnic state where one ethnic group (or nation) rules over the other, usually conquered, ethnic groups. (This applies to the Russian, British, Osman, Austro-Hungarian, Roman, Persian, Mongol, Carolingian and many other empires.)
The size is secondary.
This definition fits the Aztec empire as well.
Upvote:6
As "empires" go, the Aztec "empire" is pretty small. It ranks 212th among large empires, with 220,000 square miles. That's about the size of two large European countries, say Germany and France, or Germany and Poland. It was also larger than any other civilization in the Americas (pre Columbus), except for the Incas.
An Emperor may be considered a "king of kings." Aztec leaders had several "kings" of smaller "city states" under them. Except for three core cities, most of these cities were conquered, rather than "home-grown." By that standard, as well as the "two country" size, the Aztecs had an empire.