I'm not sure why you want to limit what an axiom can be to two types of statement. An axiom is any unproven mathematical assumption we make, so we can regard any well-formed mathematical assertion as an axiom. Whether a statement is good axiom is a different story. For instance, we should avoid assuming things if we can prove them from our other assumptions, we should try to make the things we assume well-motivated and consistent. (Perhaps even 'true' if we have a good sense of that, or better yet, 'self-evident', if that means anything).
Many axioms do, as you suggest, assert the existence of mathematical objects with certain properties. They may also assert uniqueness. For instance the parallel postulate asserts the existence of a unique line parallel to some give line, through some given point. Most of the axioms of set theory asset the existence of certain sets defined off of others.
And yes, some axioms do define how our function and relation symbols behave. For instance a lot of the axioms of Peano arithmetic exist to tell us how addition and multiplication behave, and the axioms of an ordered field tell us the properties of the ordering relation, addition, multiplication, zero, one, etc.
Note that we embed arithmetic or algebraic structures into set theory, we no longer have these operations as primitive symbols that are talked about by our axioms. Rather, we define them in terms of our set language's membership relation $\in.$ Here we use the axioms to prove such definitions are sensible: for instance we will prove, not assume that for any $x$ and $y$ there is a unique $z$ such that $x+y=z,$ where now "$x+y=z$" is an abbreviation for a relation between sets $x$, $y$, and $z$ that represent natural numbers, and this relation is defined in the language of set theory.
Additionally, how we interpret or phrase the axioms can change "what kind of thing" we think they assert. For instance, the axiom of foundation in set theory says that any set has a $\in$-minimal element, i.e. it has an element that does not contain any of its elements. So from a certain angle it says the membership relation has certain properties. But often when we think about it we imagine it as prohibiting what kinds of sets can exist. This is just a difference in language: 'every set is well founded' and 'there are no non-well-founded sets' clearly mean the same thing, but the first seems to be a property and the second to be the negation of an existence statement. So a lot of this is just about duality between universal and existential quantifiers. As another example, an existence statement of the axiom of infinity would be 'infinite sets exists' but there is also a universal version 'it is not true that all sets are not infinite'. Doesn't exactly roll off the tongue, but it's equivalent.