It is the case that all measurements proceed via the exploitation of the natural interactions that we understand theoretically. But once the measurement is completed and the result in hand, the QM analysis of the subsequent evolution of just those systems that yielded that particular result can no longer employ the original state function (which allows for all the different possible results), but must then employ just that part of the original state function that corresponds to the particular result. This 'sudden' change in the state function used is called state function collapse. Many physicists regard this change as corresponding to nothing more than the change in the experimenter's knowledge once the result is in hand. This is the epistemological interpretation of the state function. But many regard the change as also reflecting a genuine physical change in the state of those systems that came through the measurement yielding the particular result. This is the ontological interpretation of the state function and it has many variations. Still many others hold an ontological interpretation of the state function while denying that collapse happens at all.
These latter views, which also have many versions, have given rise to various interpretations of and/or alternatives to QM that go by names such as Pilot wave, deBroglie-Bohm, Modal interpretations, Relative state, Many Worlds, Many Minds, Consistent Histories, Decoherence theoretic, Information theoretic etc. Collectively these are all called NO-Collapse theories.
The champions of real, physical collapse have also been at work creating alternative theories of their own that replace the collapse postulate by evolutions that generate collapse dynamically. These theories go by the names of their authors, Ghirardi-Rimini-Weber-Pearl, Karolahazy, Penrose, Gisin, Percival, etc. Collectively these are the Collapse theories.
The difficulty in deciding among these many and still proliferating alternatives is due to the incredible success of standard QM. All the alternatives must, at least, reproduce the corroborated results of QM while possibly allowing for deviations in, as yet, untested waters. Some of them offer no deviations from QM, whatsoever! So deciding between them
and QM must be a matter of philosophy or aesthetics. In any case, the days of the hegemony of the Copenhagen interpretation, if they ever really existed, are gone forever.