I think that the following axioms describes what modern set theory is all about (on top of mono-sorted first order logic with equality and membership)
Extensionality: Two sets with the same elements are equal: $$\forall X \, \forall Y \, \big{(}\forall z \, (z \in X \Leftrightarrow z \in Y) \implies X=Y \big )$$
Replacement: if $f$ is a function definable in the langauge of set theory without symbol $``B"$, then: $$\forall \vec{P} \ \big{(}\forall A\, \exists B: B=\{f(x)| x \in A\}\big )$$
Hierarchy: Every set belongs to a limit stage of the cumulative hierarchy.
$$\forall X \, \exists \alpha \, (\lim(\alpha) \land X \in V_\alpha)$$
Order: Every set is injective to some ordinal. $$\forall x \exists \, ord \, \alpha: \exists f \, (f: x \hookrightarrow \alpha)$$
Height: Since stages of the cumulative hierarchy are indexed by ordinals, then this can take the form of existence axiom for ordinals. So the general form is: $$\exists \alpha \, (ord(\alpha) \land \phi)$$ which is usually a large cardinal axiom.
I think this general form captures the standard modern approach to set theories.
I know that there are non-well founded set theories, or even choice-less large cardinal axioms, but it doesn't appear that these fit the standard line, those are usually contemplated in parallel to the standard line.
Is the above a correct capture of what's going on with modern set theories or it is an incorrect characterization? If the latter, then in what sense it is so?