Defining property of morphisms of algebraic spaces out of an étale local on source and target property of scheme morphisms. Stacks Project vs Olsson

This is about the umpteenth discrepancy of definitions in algebraic geometry. It's about recipes to define properties of morphisms of algebraic spaces out of properties of morphisms of schemes. When the morphism of algebraic spaces in question is representable the recipe is clear [SP, 025V; Ols16, 5.1.5 (ii)] (see also first paragraph of [SP, 03HA]). The recipe for not necessarily representable morphisms seems to be more subtle.

Suppose $P$ is a property of schemes that is étale local on the source and target (in a way we'll make precise later, spoiler: the meaning depends on the author). Then we want to define what it means for a morphism of algebraic spaces to have property $P$.

Here are the two definitions I know:

Definition [Ols16, 5.4.11]. Suppose $P$ is a property of morphisms of schemes that is:

1. Stable under composition.
2. Stable under arbitrary base change.
3. Étale local on the target [SP, 02KO].
4. Étale local on the source [SP, 036G].

(These four properties is what by definition Olsson refers to as “local on domain for the big étale site $C$” [Ols16, 5.1.3 (iv)].) Let $f:X\to Y$ be a morphism of algebraic spaces. We say that $f$ has $P$ if there exists étale surjective maps $v:V\to Y$ and $u:U\to X$ such that the projection $$ U\times_YV\to V $$ has property $P$ (note that $U\times_YV$ is a scheme by [Ols16, 5.1.9]).

Definition [SP, 04RD]. Let $S$ be a scheme. Let $P$ be a property of morphisms of schemes which is étale local on the source-and-target [SP, 04QZ].¹ We say a morphism $f:X\to Y$ of algebraic spaces over $S$ has property $P$ if the equivalent conditions of [SP, Lemma 03MJ] hold.

My question is:

How do these definitions compare? If $P$ is a property of morphisms of schemes that satisfies all four properties of the above Olsson's definition and is étale local on the source-and-target [SP, 04QZ], then does “having $P$” for a morphism of algebraic spaces mean the same in both definitions?

References

[SP]. The Stacks Project Authors, The Stacks Project

[Ols16]. M. Olsson, Algebraic Spaces and Stacks, American Mathematical Society, 2016

¹“Being étale local on the source-and-target” is equivalent to having the following three properties:

1. “Being étale local on the source.”
2. “Being étale local on the target.”
3. Any (equivalently, both) of the following two conditions holds:
   • “Being stable under postcomposing with étale morphisms.”
   • “Being stable under postcomposing with open immersions.”

See [SP, 04R0, 04R2].