Recall that a formally etale morphism of schemes $Y \to X$ is a morphism which has the unique right lifting property with respect to all nilpotent thickenings $Z \to W$. One salient feature of nilpotent thickenings is that they are Zariski homeomorphisms.

Definitions:

1. Say that a morphism of schemes $Z \to W$ is weakly nilpotent if the underlying map of (Zariski) topological spaces $Z \to W$ is a homeomorphism.

2. Say that a morphism of schemes $Y \to X$ is strongly formally etale if it has the unique right lifting property with respect to all weakly nilpotent maps $Z \to W$. That is, for every commutative square as below, there exists a unique diagonal filler $W \to Y$, as indicated, making the two triangles commute.

$$\require{AMScd} \begin{CD} Z @>>> Y \\ @VVV \nearrow @VVV\\ W @>>> X \end{CD}$$

By definition, then, if $Y \to X$ is strongly formally etale, then $Y \to X$ is formally etale. The converse presumably does not hold. However, etale morphisms have an additional finiteness condition (etale = formally etale + locally of finite presentation) which makes me hope for an affirmative answer to the first question below:

Questions:

1. Let $Y \to X$ be an etale morphism. Then is $Y \to X$ strongly formally etale?

2. Does there exist standard terminology for "weakly nilpotent" / "strongly formally etale"?

3. Is there a characterization of the class of morphisms which have the unique left lifting property with respect to all etale morphisms? How about (strongly) formally etale?

EDIT: Thanks to Harry Gindi and Anonymous below for insightful comments, I've refined the definitions here. Recall that a formally etale morphism of schemes $Y \to X$ is a morphism which has the unique right lifting property with respect to all nilpotent thickenings $Z \to W$. One salient feature of nilpotent thickenings is that they are universal homeomorphisms.

Definition: Say that a morphism of schemes $Y \to X$ is strongly formally etale if it has the unique right lifting property with respect to all universal homeomorphisms $Z \to W$. That is, for every commutative square as below, there exists a unique diagonal filler $W \to Y$, as indicated, making the two triangles commute.

$$\require{AMScd} \begin{CD} Z @>>> Y \\ @VVV \nearrow @VVV\\ W @>>> X \end{CD}$$

By definition, then, if $Y \to X$ is strongly formally etale, then $Y \to X$ is formally etale. The converse presumably does not hold. However, etale morphisms have an additional finiteness condition (etale = formally etale + locally of finite presentation) which makes me hope for an affirmative answer to the first question below:

Questions:

1. Let $Y \to X$ be an etale morphism. Then is $Y \to X$ strongly formally etale?

2. Does there exist standard terminology for "strongly formally etale"?

3. Is there a characterization of the class of morphisms which have the unique left lifting property with respect to all etale morphisms? How about (strongly) formally etale?