What is the exact nomenclature of ionization? I'm a little confused as to what reactions are considered ionizations. Is there any requirement for the educts, for example? 

Some references suggest that it's only considered an ionization if you produce additional charged atoms/molecules from the educts (and some processes are dubbed detachment, which intuitively look like ionizations to me). Then "double ionization" wouldn't be possible, strictly speaking. To what extend have electrons to be produced. At the moment I think that it's legal to call most processes that look like ionization ionization.

There is a remark in the wikipedia article regarding the distiction from dissociation, but I don't really understand it.
The reference for chemical terminology is the IUPAC Gold Book, or IUPAC Compendium of Chemical Terminology. The entry for ionization is as follows:


  The generation of one or more ions. It may occur, e.g. by loss of an electron from a neutral molecular entity, by the unimolecular heterolysis of such an entity into two or more ions, or by a heterolytic substitution reaction involving neutral molecules […] The loss of an electron from a singly, doubly, etc. charged cation is called second, third, etc. ionization. This terminology is used especially in mass spectroscopy.


It includes examples that you might find instructive. It confirms your intuition that the term has a broad understanding, covering a wide variety of reactions.