The authors introduce the notion of 1-harmonic-Killing (1-h-K) vector fields, i.e., vector fields whose corresponding 1-parameter group of local transformations consists of maps which have vanishing linear part of their tension field. It is shown that a vector field is a Jacobi field along the identity map if and only if it is a 1-h-K vector field. Examples of 1-h-K vector fields on (co-)Kähler manifolds are provided by the following result: for a given 1-h-K vector field on a (semi-)Riemannian manifold \((M,g)\) and a given parallel (1,1)-tensor field \(T\), the vector field \(TX\) is 1-h-K iff \(T\) commutes with the Ricci operator of \((M,g)\). Further, the authors introduce the notion of harmonic-Killing vector fields, i.e., of vector fields, whose 1-parameter group of local transformations consists of harmonic maps. These are characterized as Jacobi vector fields with harmonic flows. The relations between Killing, affine-Killing, conformal and harmonic-Killing vector fields are described, and the characterization of these kinds of vector fields with respect to their corresponding sections is provided.