What is the exact nomenclature of ionization?
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.