The following is the definition of infinitesimal generator from Oksendal.
Let $\{X_t,t\in[0,T]\}$ be a time-homogeneous It\^o diffusion in $\mathbb{R}^d$. The $\textit{infinitesimal generator}$ $\mathscr{A}$ of $X_t$ is defined by $\mathscr{A}f(x)=\lim_{t\downarrow 0}\frac{\mathbb{E}_x[f(X_t)]-f(x)}{t},\,\,\,\,x\in\mathbb{R}^d$ The set of functions $f:\mathbb{R}^n\rightarrow\mathbb{R}$ such that the limit exists at $x$ is denoted by $\mathscr{D}_A(x)$, while $\mathscr{D}_A$ denotes the set of functions for which the limit exists for all $x\in\mathbb{R}^d$.
But there is another definition of infinitesimal generator defined by semi-group, which works on the both time-inhomogeneous and time-homogeneous process.
So are those two really equivalent to each other?