Expressing "does not imply''
The formalization of a sentence in ordinary discourse claiming that "$A$ implies $B$" or "$A$ does not imply $B$" is outside the languages of propositional or first-order or higher-order logic. It is in the language of logical consequences, which roughly consists of pairs of (sets of)$-$propositional of first-order or higher-order$-$formulas.
When we use the expression "$A$ does not imply $B$" in ordinary discourse, we can mean two distinct things:
$\models \lnot (A \to B)$, which means that $\lnot (A \to B)$ is a valid (or provable) formula. This amounts to say $\models A \land \lnot B$, i.e. every structure satisfies $A$ but not $B$.
$A \not \models B$, which means that $B$ is not a logical consequence of (or is not provable from) $A$, so there exists a structure where $A$ is true and $B$ is false. This amounts to say $\not \models A \to B$.
The difference is essentially whether we mean "not" as part of the language of propositional logic, or as a negation of the relation of logical consequence. Usually, the intended meaning is 2.