Let $(M,g)$ be a Riemannian manifold and let $\mathrm{Diff}(M)$ denote the space of all diffeomorphisms of $M$.
If one considers a map $\Phi:(−\epsilon,\epsilon)\to\mathrm{Diff}(M)$ such that $\Phi_0:=\Phi(0)$ is the identity, then we can define a vector field $X(p) = \frac{\partial}{\partial t}|_{t=0} \Phi_t(p)$ for all $p\in M$. More generally, we can define $X_t(\Phi_t(p)) = \frac{\partial}{\partial t}\Phi_t(p)$ so that $X_0 = X$. We abbreviate this by writing $X_t = \frac{\partial}{\partial t}\Phi_t$. We often refer to the family $\Phi_t$ as a one-parameter family of deformations and $X_t$ as its first-order deformation vector field.
I'm curious about the equation $$X_t(\Phi_t(p)) = \frac{\partial}{\partial t}\Phi_t(p).\tag{1}$$ Why does this equation make $X_t$ a vector field? Why don't we define $X_t(p)$ at each $p\in M$ as we do to $X$? I mean, why can we not define $X_t$ by $$X_t(p) = \frac{\partial}{\partial t}\Phi_t(p)?$$ In the definition of $X$, the map $t\mapsto \Phi_t(p)$ can be seen as a curve in $M$, so it makes sense to talk about the velocity of $t\mapsto \Phi_t(p)$ at $t=0$ and define $X(p)$ to be this velocity, but I don't see how (1) works to define another vector field $X_t$. And, what is a first-order deformation vector field for? I googled the internet but didn't find any threads talking about this term. I need some help, please. Thank you.