What even are pure and applied math, anyway?

Not really a serious post.

I see the words "pure math" and "applied math" used a lot, and there seem to be some completely distinct meanings of the phrases:

1. Formal math and informal math -- you can certainly approach things like summing divergent series completely formally (follow the link for proof!), and I'm sure you could in principle be hand-wavy with category theory. So this is really about the method with which you do mathematics, not the field itself. An example of where you see this is the distinction between analysis and calculus (well, a distinction -- sometimes calculus is defined specifically as having to do with differentials and integrals while analysis is a broader field).
2. Abstract math and concrete math -- this really has multiple levels: category theory, abstract mathematics, mathematics, science, engineering, specific numerical calculation. The line is often drawn either before or after "mathematics".
3. Theoretical and applied -- closely related to the previous point, differing by the purely social question of the purpose of the study.
4. Everything else vs statistics -- I think this arises from a conflation between statistics and applied/concrete statistics. Statistics can really be a totally formal field of mathematics or even abstract mathematics, but I guess people often fail to draw the distinction (unlike, say, between "differential equations" and "applied differential equations in engineering").
5. Algebra vs everything else -- Perhaps a result of the fact that analysis and geometry often restrict to handling special concrete objects like the real and complex numbers.

I guess the reason these distinctions are often taken as synonymous is that they're quite correlated. As you get more abstract, you may feel a stronger obligation to be more formal to make sure you haven't missed out on some so-called pathological cases (although I think it's perfectly possible to develop intuition for such pathological situations, see e.g. my e^(-1/x) article, or the topology series). When working for an applied purpose, it may not be useful to be too formal, for practical constraints.

The correlation really lines up with the fundamental "purpose of mathematics". The point of having axiomatisations is that someone applying abstract ideas in concrete situations can just check if the axioms are satisfied -- and so you really must formally deduce things from them to make sure you're not making some assumptions specific to one concrete situation that you have in mind.

(Another example of such ambiguity is the distinction between "theoretical science" and "practical science". I've still not figured out if the latter refers to experimental science or applied science, and there isn't even any correlation between the ideas here.)

Why are calculus and linear algebra taught early?

Linear algebra and function theory are related — you can construct plenty of accurate analogies here, like functions and vectors, linear transforms and integral transforms, etc. In addition, the elementary techniques of calculus allow you to talk about non-linear transformations in a pretty nice manner — e.g. the Jacobian matrix as a change-of-basis matrix for non-linear co-ordinate transformations.

In general, calculus is just a special case and a “constructivist” kind of way of understanding the much deeper mathematical field of analysis. The calculus of variations, basic complex analysis, matrix calculus, etc. are other examples of this. It’s taught, despite its non-fundamental nature, not only because it locally linearises things with infinitesimals, allowing us to study non-linear things, e.g. in differential geometry, but also because a lot of its results are special cases of purer results in advanced mathematics. Some elementary examples: the chain rule, a special case of a change-in-basis-variables/the Jacobian matrix; the fundamental theorem of calculus and Stokes’ theorem, special cases of the generalised Stokes’ theorem in differential geometry.

Linear algebra is taught for similar reasons — it introduces you to a lot of things in algebra, much like how calculus introduces you to a lot of things in analysis. Together, they also introduce you to a lot of things in geometry — largely because the “ideas” behind the two allow us to describe a lot of things in a linear way — completing the algebra-analysis-geometry trinity.