A Killing vector on a (pseudo-)Riemannian manifold is equivalently
a covariantly constant vector field : a vector field $v \in \Gamma(T C)$ that is annihilated by (the symmetrization of) the covariant derivative of the corresponding Levi-Civita connection;
Similarly a Killing spinor is a covariantly constant spinor.
For $(X,g)$ a Riemannian manifold (or pseudo-Riemannian manifold) a vector field $v \in \Gamma(T X)$ is called a Killing vector field if it generates isometries of the metric $g$. More precisely, if, equivalently
the Lie derivative of $g$ along $v$ vanishes: $\mathcal{L}_v g = 0$;
the flow $\exp(v) : X \times \mathbb{R} \to X$ is a flow by isometries.
The flows of Killing vectors are isometries of the Riemannian manifold onto itself.