Why is the Copenhagen interpretation the most accepted one? I would say the answer is this:
- it's the oldest more or less "complete" interpretation
- hence you'll find it in many (all?) early text books, which is basically from where people writing modern text books copy from.
- the overwhelming majority of physicists doesn't really care about the interpretation, since it (up to now) is only a matter of philosophy. We cannot know what interpretation is correct, because we can't measure differences, hence the interpretation question is a matter of taste rather than scientific knowledge.
- most standard QM courses at university (at least the ones I know) don't bother with the interpretation. They just introduce the concepts, updates of knowledge, etc. and in that sense, the Copenhagen interpretation is just convenient.
This implies that if you ask a lot of physicists, some have never even thought about the matter.
If interpretation is a matter of philosophy, why should we worry about it then? I can think of two points here:
a) By thinking also about interpretations of our theory we may come up with new theories that give us "nicer" interpretations of existing results, but they are essentially inequivalent to quantum mechanics. Bohmian mechanics from what little I understand about it is such a candidate, which might turn out to at one point make different predictions than classical quantum mechanics (up till now, it's just a different interpretation). This is of course a very good reason to think about it, because if quantum mechanics can not explain everything and there is a better theory, which can explain more with similarly "simple" assumptions, we want to have it.
b) It might help our understanding of "reality". This is only interesting, if you believe that your theory describes reality. If you believe that we only ever create effective models that are limited to a certain domain of our variables, then interpretations become uninteresting. Your model isn't the real deal after all, so why bother with something, you can't measure? It doesn't enhance our knowledge.
So, if you don't believe that science should (or even can) provide ontologic theories and if you don't think a better theory than quantum mechanics is maybe just beyond the horizon, then you don't care about interpretations of quantum mechanics. Otherwise, you should.
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.
Best Answer
At first a big warning: Do not rely on stuff that you find on the Internet without further reference, there are many, many people, even recognized physics professors, who dont have a clue what they are talking about. Also, I would not recommend Deutsch's writings about MWI until you have a good understanding of the basics, until then, they are rather confusing.
For an elementary Introduction I would recommend the Stanford Encyclopedia of Philosophy (http://plato.stanford.edu/index.html), it contains good introductions to the various Interpretations such as MWI or CI.
Furthermore, a classic on this topic is "Jammer - The Philosophy of Quantum Mechanics", a very detailed description of the historical development and the different interpretations. If you want a detailed and precise overview over the relevant Interpretations, this book is for you, although its partially not too easy to read.
Apart from handpicked books, I would mainly recommend reading the original papers, especially the one by Everett, who is often cited very, very wrong. "The Theory of the Universal Wave Function" is even available online, i guess (must-read for everyone who is interested in Interpretation stuff). Apart from that, the most comprehensive collection of the original articles can be found in "Wheeler, Zureck(ed.) - Quantum theory and measurement". It contains most of the historically relevant articles. Surprisingly, they are mostly quite easy to read, the math does not exceed the level from elementary Quantum Mechanics.
Good luck, we need more people who know what they are talking about when it comes to interpretation!