In Wedhorn's and Görtz's book, a morphism $f: X \to Y$ is said to be smooth of relative dimension $d$ when for every $x \in X$, there exists affine open neighborhoods $U$ of $x$ and $V = \operatorname{Spec} R$ of $y=f(x)$ such that $f(U) \subseteq V$, and an open immersion of $R$-schemes $j: U \hookrightarrow \operatorname{Spec} R[T_1, \dots, T_n]/(g_1, \dots, g_{n-d})$ such that the matrix $\left (\frac{\partial g_i}{\partial T_j}(x)\right )$ has rank $n-d$.
In Bosch's book, $f$ is smooth of relative dimension $d$ if for every $x \in X$, there exists an open neighborhood $U$ of $x$ and an $Y$-morphism $j:U \to W \subseteq \mathbb{A}_Y^n$ (here, $\mathbb{A}^n_Y = \underline{\operatorname{Spec}}_Y \mathscr{O}_Y[T_1, \dots, T_n]$) giving rise to a closed immersion from $U$ over an open subscheme $W \subseteq \mathbb{A}^n_Y$ satisfying:
If $\mathcal{I} \subseteq \mathscr{O}_W$ is the ideal correspondent to $j$, then there are $n-d$ sections $g_1, \dots g_{n-d}$ generating $\mathcal{I}$ in a neighborhood of $z = j(x)$ and such that the matrix $\left(\frac{\partial g_i}{\partial t_j}(z)\right)$ has rank $n-d$, where $t_i$ are the coordinate functions of $\mathbb{A}^n_Y$.
To be honest, I can't quite make sense of Bosch's definition: what does $\frac{\partial g_i}{\partial t_j}$ even mean, given that the $t_j$ are global sections and the $g_i$ are sections on a neighborhood of $z$, and, thus, not necessarily polynomials. Am I missing something? How can I prove that these two definitions are equivalent? I can sort of see how Bosch's definition imply Wedhorn's and Görtz's, if I squint. Can someone help me?
Question: "To be honest, I can't quite make sense of Bosch's definition: what does $∂g_i/∂t_j$ even mean, given that the $t_j$ are global sections and the $g_i$ are sections on a neighborhood of $z$, and, thus, not necessarily polynomials."
Answer: You can choose $x\in U:=Spec(B)\subseteq X$ and $V:=Spec(A) \subseteq Y$, open affine sets with $f(U) \subseteq V$ such that the restriction $f_U$ to $U$ factors
$$j:U \rightarrow W \rightarrow^p V$$
where $W:=Spec(A[t_1,..,t_n]/I)$ and $p$ is the canonical map. The elements $t_j$ are generators of the polynomial ring $A[t_1,..,t_n]$. This is discussed in Mumford, "The red book..", page 436 (this is the old version from 1967 with 442 pages)
Note: There are several version of the book available - look for "flat and smooth morphisms" - it should not be difficult to locate this chapter.