"Aztec" as the literal meaning of the term in Nahuatl just means "Person from Aztlan", which is the legendary place the Mexica and some other subgroups of the Nahua culture migrated from prior to arriving in Central Mexico. In 16th century Nahua sources youi have those documents saying stuff like "from now on you are not Aztec, but are now Mexca/Xochimilca/Tlaxcalateca/etc"
But the term Aztec as we use it dates to like 18th/19th century anthropological stuff where the term was re-adopted. I forget for what specific definition, but the bottom line i that, today, "Aztec" gets used very inconsistently, even by reliable sources. Depending on what article or book you are reading, it could mean anything from:
- The Mexica speffically.
- The broader Nahua culture/civilization
- The cities of the Triple Alliance (Tenochtitlan, Texcoco, and Tlacopan)
- The Triple Alliance itself + it's subject states (The "Aztec Empire")
This obviously gets confusing, and you sort of just need to figure out via context which definition a given source is using. Also, because not all Nahuas are Mexica (there are other subgroups like Acolhua, Tepaneca, Xochimilca, Chalca, Tlaxcalateca, etc), not all triple allliance citites are Mexica (Texcoco is Acolhua and Tlacopan is Tepaneca); at certain points in time not all Mexica are in the Triple alliance (when Tlatelolco split off from Tenochtitlan); not all cities in the Aztec empire were nahua (many conquered cities were Zapotec, Mixtec, Otomi, Huastec, etc) not all nahua states were inside the Aztec Empire (Tlaxcala was Nahua (Tlaxcalateca), but was never conquered), etc.
Generally speaking, I try to just say Mexica, Nahua, etc as relevenant and use "Aztec" exclsuively to mean or relate to the "Aztec Empire" as a state or expansionistic efforts, but sometimes I forget to explain all this and just use "Aztec" generically to also mean the Mexica speffically or Nahua cultural stuff in general.