From what I understand, there are these different notions of exponentiating a "vector":
The normal exponential map in Riemannian geometry - $\exp_p({\bf v})$ is the point $\gamma(1)$ along a geodesic $\gamma$ such that $\gamma(0) = p$ and $\gamma'(0) = \bf v$.
In matrix Lie groups, the Lie algebra can be thought of as "vectors" (tangent vectors to the Lie group at the identity), and the exponentiation is the matrix exponential.
Thinking of "vectors" as a differential operator, e.g. $\partial_x$, then exponentiating such an operator yields a "shift" operator of some kind.
I can accept that the first two notions of exponentiation coincide for a particular metric (from the Killing form) on the Lie group, but I am wondering whether the third notion is relevant, or completely unrelated.