I find the definition of conjugate points on Wiki
Suppose $p$ and $q$ are points on a Riemannian manifold, and $\gamma$ is a geodesic that connects $p$ and $q$. Then $p$ and $q$ are conjugate points along $\gamma$ if there exists a non-zero Jacobi field along $\gamma$ that vanishes at $p$ and $q$.
I notice that, for defining a pair of conjugate points, we need to choose a geodesic. So can we define the conjugate points using the following method?
Suppose $p$ and $q$ are points on a Riemannian manifold. Then $p$ and $q$ are conjugate points if there exist a geodesic $\gamma$ that connects $p$ and $q$ and a non-zero Jacobi field along $\gamma$ that vanishes at $p$ and $q$.
or
Suppose $p$ and $q$ are points on a Riemannian manifold. Then $p$ and $q$ are conjugate points if for any geodesic $\gamma$ that connects $p$ and $q$ there exists a non-zero Jacobi field along $\gamma$ that vanishes at $p$ and $q$.
Are these two definitions above equivalent? If not, what are the counterexamples? That is, can we find a Riemannian manifold $M$, two points $p$ and $q$ on $M$, and two different geodesics that connect $p$ and $q$ such that $p$ and $q$ are conjugate along one and non conjugate along another.