In this context, I believe we are to consider $\exp(t,p)$ as a map $\exp_t : S \to M$, and take a regular value to be a point $q \in \exp_t(S)$ such that the push-forward $\exp_{t*} : T_{\exp_t^{-1}(q)}S \to T_qM$ is a surjection. Here $t$ is to be considered merely a parameter which tells us which map we are dealing with. In other words, $q$ is a regular value we would have $T_qM = \exp_{t*}(T_{\exp_t^{-1}(q)}S).$ Since $S$ has co-dimension one, this is obviously only possible if the pre-image $\exp_t^{-1}(q)$ contains more than one point. At least this would recover, in some sense, the intuitive notion of conjugate points, and bears some resemblence to the standard definition.

In this context, I believe we are to consider $\exp(t,p)$ as a family of maps $\exp_t : S \to M$, and take a regular value to be a point $q \in \exp_t(S)$ such that the push-forward $\exp_{t*} : T_{\exp_t^{-1}(q)}S \to T_qM$ is a surjection. Here $t$ is to be considered merely a parameter which tells us which map we are dealing with. In other words, $q$ is a regular value we would have $T_qM = \exp_{t*}(T_{\exp_t^{-1}(q)}S).$ Since $S$ has co-dimension one, this is obviously only possible if the pre-image $\exp_t^{-1}(q)$ contains more than one point. At least this would recover, in some sense, the intuitive notion of conjugate points, and bears some resemblence to the standard definition.