Min-cut in a network with zero flow from source to sink
No, your gut feeling is not correct.

Consider the following flow network with source $s$ and sink $t$, where the capacity of every edge is 1. The max-flow from source to sink is 0. The s-t cut $(\{s,A\}, \{B,t\})$ is a minimum cut since the only connecting edge $(B, A)$ goes from sink side to source side.

$$ s \longrightarrow A \longleftarrow B \longrightarrow t$$

You could learn several lessons from this simple problem.  One is that source and sink are symmetric in some strong sense. It looks like your gut feeling does not respect this symmetry. Another lesson is that a few simple examples should be included in your repository. Always try constructing some simple or extreme examples when you want to check a proposition of unknown truth value.