Ordinary talk is full of ambiguity and vagueness.
Mathematics tends towards being much more precise and rigorous, with many terms explicitly defined. Sometimes these terms are made up for mathematical use, such as “homomorphism.” Other times, they are terms that already have some meaning (or several meanings) in everyday talk, but in mathematics they are co-opted for a specific, explicit purpose, that may be only loosely inspired by the everyday meaning. Some examples of this are words like “lattice” and “compact.”
In everyday talk a “lattice” might be a fence pattern like this. In mathematics it has more specific meanings.
However, although mathematics is definitely more precise and explicit than ordinary, everyday talk, it still ends up being messy. This might not be surprising, given how many different people contribute to mathematics over so many years.
One kind of messiness is that the same words can be used in different parts of mathematics to mean different things. For example, the word “lattice” means one thing if you’re talking about mathematical orders and/or algebras, and something different if you’re talking about mathematical groups. The word “compact” means one thing if you’re talking about topological spaces, and something different if you’re talking about logical theories. And so on.
Another kind of messiness is that sometimes mathematicians define a word in different ways, even when they intend it to have the same meaning, in some sense. I said that one way to use “lattice” is to talk about mathematical orders and/or algebras. Even for this one intended use, there are different definitions offered.
The details aren’t important, but if you’re curious, here is one definition and here is another.
This can sometimes be harmless, if the two definitions are after all equivalent. But sometimes it happens that that the mathematicians start out thinking that some definitions are equivalent, but then later it turns out that they’re not. Or it might turn out that if we make some assumptions, they are equivalent, but on other assumptions they’re not. And those assumptions might be controversial: accepted by some theories but rejected by others.
When it comes to definitions, philosophy often tends towards the mathematical model. Sometimes it’s just as explicit and precise as mathematics, sometimes it’s just loosely headed in that direction. This can often be philosophically useful.
It’s important to realize though that the kinds of messiness described in mathematics all show up in philosophy too, even more often and making more of a mess.
Sometimes one and the same word will be used in different parts of philosophy with different meanings. Prominent example of this are words like “internal” and “external”.
And even when we focus on one intended meaning, in some sense, sometimes different philosophers (or even the same philosopher in different places) will define the word in different ways. Sometimes these differences are minor subtleties, that might matter for some purposes but not others. Other times these differences can be pretty substantial. Usually the philosophers offering the definitions expect that they are equivalent, and choose one of them rather than the other because it’s easier to work with in the discussion they’re engaged in. But often these expectations of equivalence are controversial. Often it’ll be the case that if we make some assumptions, the definitions are equivalent, but on other assumptions they’re not. And those assumptions might be accepted by some theories but disputed by others.
Sorry about all this. It does makes philosophy harder to learn. But I think it’s better to acknowledge it, and help you see where it’s happening, than to paper it over and pretend it’s not going on.
The alternative would be for your teachers to choose one of the definitions, either the one they like best, or that’s used in the textbook they’re using, and just declare that this is the official definition of a term like “substance”, or “physicalism”, or “reduction,” or what have you. But then sooner or later you’re going to come across some other philosopher working with a different “official” definition. And it can then be confusing what’s going on. Until you get enough experience to figure out what I’ve told you above.