Proving that a property is k-inductive with an SMT solver (parametric resettable counter)
I don't see any gap in your understanding.  What you wrote looks correct to me.

Perhaps the author meant to use the property $P := c \le n+1 \land n \ge 1$.  I think with that modification, $P$ becomes 1-inductive but not 0-inductive.  I suspect the author was looking for a simple example of a property that is 1-inductive but not 0-inductive, and missed a small detail.

Good job catching this!