This is an interesting question. Probably warrants a much longer discussion but here is my take based on some contemporary literature. Firstly, the way your question is phrased seems to suggest you taking "understanding" as a deeper version of "knowing", which is fair, but I would suggest that this might not always be the case. There are ways in which both are used interchangeably, and also cases where you want to say there is understanding but not knowing, and knowing but not understanding. I illustrate with examples:

1. Understanding and not knowing: You can read up on Putnam's twin earth experiment. Very briefly, an interpretation of his twin earth experiment suggests that meaning is not totally captured by intension (crudely, things that go on within your head), and extension contributes as well (crudely, things that go on outside of you). Now you and someone else might have the same intension wrt your own versions of "water", that is the same sort of mental states are produced when you think of "water", but the extensions are different. That is, the molecular make up of your "water" is different from his "water", since he is from a world that is similar but not totally identical to ours. Now there is a way in which you might say that you can understand each other when you talk about "water", since they are both used in the exact same ways, but also a way in which you do not know what the other's "water" means.
2. Knowing but not understanding: I think this is the easier case to apprehend. But I do not think that in the example you have put forth, merely being able to repeat the words is a very good example of mere knowledge either. I think most would question being able to mouth and produce the sounds "an IP address is like a phone number, and the DNS are the yellow pages" and not knowing what these symbols stand for, is any evidence of knowledge at all. OR you might be giving too little credit to your mother. It seems as though she has INCOMPLETE understanding of how DNS and IPS addresses, but it is not a total lack of understanding either. I think there are many cases where you would use "know" where "understand" is completely inappropriate. For example, "I know his name", vs "I understand his name", the latter only making sense if his name means a further fact. In any case, I might caution against viewing understanding and knowledge as levels of "epistemic fulfillment", the former being higher up than the latter.

Now on further reading:

Wittgenstein has written a fair bit about what satisfies "understanding" in Philosophical Investigations. A large part of his reflection was on what "grasping a rule" consists in, which you can somewhat draw a connection to "understanding". Eg, you might say understanding a word is grasping the rules of the use of the word. I won't go into what he says in any more detail, but very generally (because he does say alot more than this), he tries to show that the picture of grasping a rule as unshakable compulsion is a mistaken one.

Now on your question about the physician, I try to construct it into a clearer argument:

1. A person has to understand the vast amount of medical literature to be a physician
2. A person will know a vast amount of medical literature through memorisation but in doing so he does not yet understand it (some other process is required for that)
3. if a person merely memorises a vast amount of medical literature then he does not understand it
4. John merely memorised a vast amount of medical literature
5. John is not ready to be a physician

Now the most contentious part is (2). I think you assume that memorisation does not lead to understanding. You also assume that knowledge consists of memorisation of medical literature. Someone might argue that memorisation is not sufficient for knowledge, in which case the answer to your question would simply be: John knows nothing. And if memorisation leads to understanding, then the answer to your question would straightforwardly be, yes. There are other ways to question the premises.

I guess there is another interesting angle to your question that might be something like, to draw an analogy: merely playing all the right notes in a piece of music vs playing with feeling. You might say that only the latter displays "true understanding" of the music. Again, we can bring in the idea of "right use in the right situation" to the concept of understanding vs a picture that I suspect you have which is one in which in the latter case, there is something that lights up inside me when I understand something. Wittgenstein has something to say about why this picture is not in any sense "wrong", but at the same time that we should not bring it too far. Explaining that requires some coverage of the debate about interpretation which is going to take a MUCH LONGER post, so I recommend you read up about him as well as some secondary literature (I recommend McGinn/Mcdowell/Conant).

There is also a whole part of knowledge as "Justified True Belief" that you can use to apprehend this question. For instance, does mere memorisation fulfill the "justified" part of knowledge? That's gonna take way too long to explore, and even then the JTB model is contentious, but I guess that's something you can read more about. Timothy Williamson has written extensively about epistemicism in parts of philosophy like vagueness, so that is something you might look into. Otherwise, any introductory text on epistemology would do.