Logical constraint in ILP

The big-M constraint $z le M t$ does enforce $z > 0 implies t = 1$, equivalently its contrapositive $t = 0 implies z = 0$, but not the converse $$z = 0 implies t = 0. tag1$$ To enforce $(1)$, consider its contrapositive $$t = 1 implies z > 0 tag2,$$ which you can enforce via big-M constraint $$epsilon - z le (epsilon - 0)(1 - t),$$ equivalently, $$z ge epsilon t,$$ where $epsilon > 0$ is a tolerance that represents the smallest value of $z$ that you would consider to be positive.