I have a general question about the motivation behind to definition the smooth morphisms as we know it from algebraic geometry. The most common definition of a smooth morphism $: X \to Y$ between two smooth Noetherian schemes $X,Y$ is:

$f$ is smooth if and only if

(i) $f$ is flat and locally of finite presentation

(ii) for every $y \in Y$ the fiber $X \times_Y k(y)$ is smooth variety over $k(y)$

I read recently that the motivation of smooth morphisms in algebraic geometry arise as a kind of imitation attempt of a maps called "submersions" in differential geometry. Namely if $X,Y$ are smooth manifolds then a submerison $s: X \to Y$ is a surjective, proper $C^{\infty}$ map for which for every $x \in X$ the induced differention $D_xf: T_x X \to T_y Y$ is surjective. The Ehresmann's lemma says that such submersion is moreover a locally trivial fibration. That seems to coincide with the intuition that flaness is something like a continuous family of neighbored fibers.

My Question is if there exist a definition of a smooth morphism $: X \to Y$ in algebraic geometry world, which emphesize more immediately that the motivation for smoothness in algeom arise from submersions in differential geometry?

Let look again at definition above. (ii) seems reasonable, since this tells that every fiber of $f$ is smooth, ie morally a manifold. But that reason that the point (i) arise immediately from the differential geometry isn't immediately clear if one not believes that flatness makes families "continuous". Morally "continuity of fibers" (= flatness) should be a consequence (like by of Ehresman lemma in differential geo), not an immediate "part" of definition.

Can the flatness requirement be replaced in algeobraic definition by requirement that the induced differention $D_xf: T_x X \to T_y Y$ is surjective? And is this equivalent of flatness in algebraic setting?

The reason is that I conjecture that this could be true is that we can surely
define the tangent space of $X$ at every $x$ *pure algebraically* as the dual
of the stalk $\Omega_{X,x}$ or equivalently as $\{ \phi \in \operatorname{Hom}
(\operatorname{Spec}k[\epsilon], X) \ \vert \text{ Im } \phi = \{x\} \}$.

The Question is if in algebraic setting the surjectivity of algebraic $D_xf: T_x X \to T_y Y$ at every $x$ is equivalent to flatness of $f$?