There is a strong sense within the recognised set of sciences, that what really determines the boundaries are the points where emergent phenomena become more relevant than their sources.
There is a level where the actual locations of hydrogen ions are strictly relevant, and one where the notion of acidity takes over. There is a level where Brownian motion is a relevant phenomenon, and a boundary beyond which where temperature is a more useful concept that means the same thing in a more abstract sense. There is a level where cells are important and a level where tissues and organs are relevant, and then another where all that matters are whole bodies.
Of course, these boundaries do not align well. We know that in basic chemistry we are going to attend to acidity and temperature, rather than ion positions and Brownian motion. But addressing different specialities within the discipline of chemistry, say crystallization or the details of polar vs nonpolar solvents, we might be using one of the two more emergent concepts and one of the two more basic ones.
But, in general, we can separate out towers of increasingly emergent disciplines, for example from nuclear physics, to chemistry, to biology, to psychology, to sociology, in a way that makes sense. Of course then even in that one string, there are lots of interior details like microbiology and a few other additional layers between chemistry and biology, and things like animal behaviorism lying between biology and psychology.
And obviously this is not a set of strictly linear successions. It branches. Psychology in particular lacks coherence and forks into general, personal and 'rational' psychology (among others) and will have sociology as its natural successor in one direction, while in two others it can be seen as basic to anthropology and ethnology, or to humanistic disciplines like history and literature.
But as a general way of deciding which science you are doing, you can rely upon how many and which among the concepts basic to your thinking are actually emergent phenomena too abstract to make strict sense in other sciences.
I would push this sense to its logical extreme.
I think the same pattern applies to a great degree outside of science. Philosophy's great divisions, for instance tend to cohere largely around what you are willing to treat as emergent concepts, and what you are going to attend to at a level of greater detail. Ethics is not free of logic or epistemology, but it takes basic understanding and motivation as an emergent phenomenon that relies upon them. Politics depends upon feelings and intuitions that arise out of personal ethics as emergent phenomena. And a similar network of chosen levels of attention can map out the rest of the subdisciplines.
Some of these boundaries are much more contentious, but can really be seen in the same light. One way of looking at those things is that they are places where some things are deeply emergent, and others are very low-level at the same time.
The very contentious boundary you bring up first, where mathematics meets advanced physics like relativity and quantum mechanics can be seen as a place where we depend deeply upon mathematics, as emergent from psychology and logic, to tell us what can be reasonably understood and communicated, while stripping away a lot of layers of emergent concepts and focussing on absolute simplicity in every other sense.