It is neither the literal title of empire (Portugal was a colonial empire, but had the title of a kingdom),
nor ruling over multiple ethnicities (the German Empire only ruled over Germans, with just a small Polish and Danish minority),
nor ruling over multiple consituent states (the Second French Empire only ever ruled over the French state, the old medieval French states like the Duchy of Aquitaine or Burgundy were long gone, meanwhile the Austrian Empire ruled over many constituent states like the Kingdom of Hungary and the Kingdom of Lombardo-Venetia, but both were empires)
nor is it owning colonies, as many empires did not own colonies (Swedish Empire)
Nor is it economically influencing other states (Soviet Union, USA) as essentially every nation would be an empire then.
Just to clarify, the Second French Empire, whether we mean the reign of Napoleon III or the return to colonial expansion that began in 1830, did include multiple constituent states. The one I am most familiar with was the Kingdom of Tahiti which was a French protectorate during the reign of Napoleon III and, Raymond Betts argues, the model upon which future protectorates were based in the empire. The treaty of annexation only occurred in 1880 after Pomare V abdicated his throne. Even in the absence of this consideration, it was a colonial empire very much engaged in expansion under the reign of Napoleon III.
As well, the very nature of the German Empire as a colonial empire would, at least to me, would naturally imply that the empire did rule over multiple ethnicities. (I realize someone implies on wikipedia's "Empire" page that Germany fits in the notion of empire only "semantically" and not because it had "additional territory or hegemony" but that's not correct.)
I am not sure the use of the word is any more complex than Merriam-Webster includes in their online edition.
(1a.1) a major political unit having a territory of great extent or a number of territories or peoples under a single sovereign authority
The use of "empire" by historians can vary a little bit between self-descriptive: those bodies that described themselves as or used the terminology of empire; and then descriptive: those bodies that match the above definition, essentially political and territorial expansion. The use of empire is not always straightforward and not always agreed upon, but most of the examples you've raised look pretty settled to me.
The examples were all engaged in territorial expansion that resulted in the incorporation of "a number of territories of peoples under a single sovereign authority." While, Sweden for instance was not engaged in colonial expansion (though it was engaged in overseas commerce and did set up factory/forts in the Atlantic at that time), it was engaged in territorial expansion so that it included a diverse set of cultures and peoples centers on the Baltic.