Is the following way of thinking about the maps $\text{Ad}$ and $\text{ad}$ correct?
Given a Lie group $G$ with Lie algebra $\mathfrak{g}$ we fix $g\in G$ to obtain a map $h\mapsto ghg^{-1}$ whose differential at $e$ is denote by $\text{Ad}_g$. We simply define $\text{Ad}:G\to GL(\mathfrak{g})$ by $\text{Ad}(g)=\text{Ad}_g$. The differential of $\text{Ad}$ at $e$ is the map $\text{ad}:\mathfrak{g}\to\mathfrak{gl}(\mathfrak{g})$. Given $X\in\mathfrak{g}$ we have that $\text{ad}(X)$ is a tangent vector at the identity on $GL(\mathfrak{g})$, a vector space, so that tangent vector can naturally be identified with a curve in $GL(\mathfrak{g})$ of the form $Id+tf$ (for small enough $t$) where $f\in\mathfrak{gl}(\mathfrak{g})$ and $Id$ is the identity map on $\mathfrak{g}$. The action of $\text{ad}(X)$ on $\mathfrak{g}$ is then $\text{ad}(X)(Y):=f(Y)$.