Where to draw the line between quantum mechanics theory and its interpretation(s)?

Interpretation is whatever people don't have to agree on to have the same accurate predictions about the observable.

Classical mechanics is empirically wrong in ways quantum mechanics isn't. For example, only quantum mechanics predicts discrete energies for atomic electrons, and discrete changes in these energies from the absorption and emission of radiation. How do you get these energies? Empirically, you measure them; theoretically, you reduce it to a calculus problem. These agree; there's no "interpretation" at work there.

Meanwhile, there are experiments you can do that vary in their results from time to time, and the frequencies of the results are, again, available both empirically and theoretically. The latter comes from the same calculus apparatus. What's that? You have a formula for something called $\psi$, whose square modulus gets us the answers we want? Great, our theory is predictive (insofar as anything probabilistic deserves that label.)

But what's this $\psi$ that crops up in both of those exercises? Well, it's not a thing classical mechanics makes claims about, or experiments detect; so whatever answer you give to that question, it amounts to an interpretation of quantum mechanics. Oh, you need $\psi$ or some alternative to get the predictions, and the predictions are right; no-one disputes either of those statements. But when you ask what these items "are", or "what they do unobserved", that's interpretation.

Get 20 QM experts in the room, each of them subscribing to a different interpretation. They'll all make the same predictions about experiments' observable outcomes. And if, in an experiment that leaves an electron's position unmeasured, one of these experts says the electron is "somewhere specific we don't know", and another says the electron is "everywhere at once", and another that it "doesn't have a location", they've found something they disagree on. It's just not an observable thing.

This doesn't mean interpretation is bunkum, or interpretations are wrong, or you shouldn't think about interpretations. (Fun fact: philosophy of physics is not limited to awkward questions about quantum mechanics.) But since your question is about where the line exists between interpretations and the rest of a QM textbook's contents, well... see the bold sentence up top.

Trust me, I understand the urge to put as little philosophy into things as possible. I do, I love me some number-crunching. But that should cut both ways, i.e. you don't want too many philosophical opinions about how little philosophy physicists should be doing either. For example, "shut up and calculate" doesn't have to mean "don't have an interpretation"; to me it means, "it's 9 am and we're predicting experimental outcomes; you can wonder what's going on 'behind the scenes' when we're at the bar". (Or vice versa!)

"Philosophy" isn't necessarily worse than "physics". It's just you can discern which is which from the fact that we know better how to get everyone on the same page for some questions than for others. Maybe that's not a bad thing. You don't have to agree the lack of an interpretative consensus is "embarrassing", but it's worth knowing that consensus is lacking.

Interpretations are here to make sense of the formalism of QM. Whatever computations and symbolic manipulations you need to do to prepare and analyse an actual experiment is QM. Whatever conceptual images you use to develop an intuition about what is going on is an interpretation.

When people say "there is no collapse", this may be an interesting point they make at some level, but the fact remains that after an actual measurement you have to update the quantum state of your system. Whether one calls this "collapse" or something else does not matter, it is a required step that you just cannot do without in any practical way - so this is part of QM, not of any interpretation.

Now the bare truth is that nobody knows what QM is really about; it is still largely a puzzle. People uncomfortable with this state of affairs push their favorite interpretation and pretend everything is settled, but it is not. Once it is settled, there will be no point in the whole idea of "interpretation". There is no "interpretation" of thermodynamics, there is no "interpretation" of Hamiltonian mechanics, because the conceptual frameworks of these theories do not hurt our brains the way QM currently does. The trouble with QM interpretations is that every single one of them is just completely unacceptable at the conceptual level for a whole class of people.

For some detailed discussions about this, I recommend Laloe, 2004 and Pablo Echenique-Robba, 2013. See also Mermin, 2009 and Landsman, 2005.

There are various forms of the "Copenhagen interpretation," but most of them share the following feature: they are not really interpretations at all. Typically, a crucial part of what an interpretation of quantum mechanics does is make ontological claims ("what is real"), which are notably absent in the formalism of quantum mechanics. True, some interpretations (e.g. QBism) explicitly don't do this, but QBism still makes a strong statement about what the wavefunction is (a state of knowledge). Copenhagen generally doesn't make such claims, so it is the "operational interpretation," if you will.

So the bottom line is: if you want to avoid interpretations, the Copenhagen "interpretation" is a good way to go. The line you are thus choosing to not cross is essentially that of making ontological claims and ascribing meaning to the wavefunction, beyond as a tool to predict measurement outcomes.