I ran across this thread and saw some heated discussion on what defines an "Empire". Specifically the posts by Thatcherist_Sybil and Chazut. https://www.reddit.com/r/worldbuilding/comments/90y53n/is_the_title_empire_appropriate/
Obviously in colloquial speech the word has two common definitions "really big kingdom" or "transoceanic colonial power" but I'm interested in how historians use the word in specific contexts. There is apparently a Marxist definition, which I think mainly refers to the colonial sort of empire but there is also the idea that only states connected with Rome are empires, which I'd never heard before.
This is a question that somewhat based itself in linguistics, and the answer depends on whether you want a prescriptivist standpoint or if you want a descriptivist standpoint, each of which has their own advantages and disadvantages. From a prescriptivist standpoint an empire is any polity that is comprised of multiple smaller polities, which administrative and executive power being centered on the dominate state, an example of this would be the Austrian empire prior to the compromise of 1867, where, while Hungary did have its own diet, Austria occupied a dominate position in their relations.
Most people use a descriptivist standpoint when speaking of empire, as descriptivism is based in the common function of language, even if not technically correct. As a result it’s not quite as precise or academic, but it’s easier to communicate the desired ideas. An example of this is the American empire.
There are some notable issues like with the Holy Roman Empire wherein it technically was one state built from many, but the level of autonomy exercised by the princes, especially as we leave the 11th century and begin to move forward, was to the point that the emperor was realistically very limited in what executive actions he could take. As such there is a large amount of debate on whether or not the HRE can be considered an empire, as Voltaire was happy to remind us of.
To answer the second part of your question, many European kingdoms were referred to as kingdoms, even after colonization, it’s descriptivist definitions that have led to us referring to many kingdoms as empires, an example being the “British empire” of King George III. The “British empire” wasn’t a thing at that time, King George III simply held titles to the Kingdom of Great Britain and the Kingdom of Ireland, as well as to some continental titles such as the electorate of Hanover ( Brunswick-Lüneburg .) the title emperor/empress wouldn’t be applied to a British monarch until 1876 with The Royal Titles Act of 1876, which styled the British monarch as the emperor or empress of India.
I apologize for the organization of this, it’s 2:30am. I will come back and reformat this later, as well as fill in any missing information. The majority of this information is based on my history/foreign language research/undergrad, but I’ll also come back through and add sources if need be.