There is the example 4.4.ii in “A First Course in Abstract Algebra” by Rotman.
If $A$ is a symmetric matrix with coefficients in $k$, define an inner product by $(v,w)=v^T\cdot A\cdot w$. The reader may prove that this is an inner product and that it is non-degenerate iff $A$ is non-singular.
Non-degenerate := $\forall v. (v,v)=0 \to v=0$. Rotman's definition of an inner product does not include “non-degenerate” and positive definiteness.
$v=\begin{bmatrix}0\\1\end{bmatrix} \land A=\begin{bmatrix}0&1\\1&0\end{bmatrix} \to (v,v)=0 \land v\neq 0$. $A$ is non-singular, self-inverse. Am I missing something?