We can state the axiom of choice ($\mathsf{AC}$) in the following way $$\forall x\Big(\neg(\varnothing\in x)\longrightarrow\exists f\big(f\text{ is a function from $x$ to $\bigcup x$ }\wedge \ \forall y\in x(f(y)\in y)\big)\Big). $$This statement is very clear and explains why we call it the axiom of choice.
There are many many useful theorems that require $\mathsf{AC}$, but many of them do not require its full power. For example, we have the ultrafilter lemma ($\mathsf{UL}$), which is much weaker than $\mathsf{AC}$. We also have dependent choice ($\mathsf{DC}$), which is stronger than $\mathsf{AC}_\omega$ and weaker than $\mathsf{AC}$.
My question is, do we know of any statement equivalent to, for example, the $\mathsf{UL}$ and that maintains the choice style? By "choice style" I mean a statement like "for any set with certain properties there exists a choice function" ($\mathsf{AC}_\omega$ is an example). I am also interested in answers to the previous question but with another (interesting, if possible) statement instead of $\mathsf{UL}$.
My intuition tells me that perhaps it is not a very relevant question because, for example, for any infinite cardinal $\kappa$, there may be a statement $\varphi$ stronger than $\mathsf{AC}_\kappa$ but weaker than $\mathsf{AC}_{\kappa^+}$ and there may not even be a statement "between" $\mathsf{AC}_\kappa$ and $\mathsf{AC}_{\kappa^+}$ with this choice style (or it is very difficult), I don't know.
