Is a vector field a section or a tangent vector? This question arose when I read two statements in a book named geometric control of mechanical system. The first statement is the following (p. 84).
$C^r$-vector field on $M$ is an element of $\Gamma^r(TM)$, i.e., a $C^r$-section of the tangent bundle of $M$.
Here, $M$ is a manifold and $\Gamma^r(TM)$ is a set of sections of $TM$. But, after that, I found the following statement (p. 173)
Let $\gamma:[a,b]\to Q$ be a $C^2$-curve. A variation of $\gamma$ is a $C^2$-map $\theta: J \times [a,b]\to Q$ with the properties
(i) $J \subset \mathbb{R}$ is an interval for which $0\in \mathrm{int}(J)$,
(ii) $\theta(0, t)=\gamma(t), \forall t\in[a, b]$,
(iii) $\theta(s, a)=\gamma(a) \land \theta(s, b)=\gamma(b), \forall s\in J$
The infinitesimal variation associated with a variation $\theta$ is the vector field along $\gamma$ given by $\delta\theta(t)=\frac{d}{ds}|_{s=0}\theta(s, t)\in T_{\gamma(t)}Q$.
According to the first statement, a vector field is a section, which is a map from $M$ to $TM$. However, according to the second statement, a vector field is an element of tangent space $T_{\gamma(t)}Q$, which means a vector field is a tangent vector. How should I interpret these two statements with consistency?
A vector field is a section of the natural projection map $TM\to M$. You should imagine choosing a tangent vector at every point smoothly.
This infinitesimal variation you mentioned is a way of giving a vector field along a smooth path (not the entire manifold). The formula you gave following $\delta\theta(t)$ is telling you which tangent vector to stick onto the point $\gamma(t)$.