“$B$ does not follow from $A$” seems different from $\lnot(A\to B)$

For 'follows from', you should not use the material implication $\to$, but rather the logical implication, for which you can use $\Rightarrow$ or $\vDash$

The difference is that $\to$ is the truth-functional operator you are familiar with, but the $\Rightarrow$ is a meta-logical symbol that claims some logical relationship between two statements.

$P \to Q$ is true in some world as long as it is not true in that world that $P$ is true and $Q$ is false.

$P \Rightarrow Q$ is true is it impossible for $P$ to be true and $Q$ to be false, i.e. that there is no world in which $P$ is true and $Q$ is false.

It is the latter that we typically mean by 'follows from': no matter what the circumstances are (i.e. no matter what world we're dealing with), if $P$ is true, then $Q$ will be true.

So, to say that $Q$ does not follow from $P$, we can write $P \not \Rightarrow Q$ ... but we can not write $\neg (P \Rightarrow Q)$, because then we are mixing up logic with meta-logic. We should simply say that 'it is not the case that $P \Rightarrow Q$.

And indeed, just because it is not the case that $P \Rightarrow Q$, does not mean anything about the truth-value of $P$ or $Q$ when evaluated in some particular world. Notably, it does not mean that $P$ is true and $Q$ is false. Moreover, if in some specific world $P$ is true, $Q$ can still be true or false. So these are all things you were looking for.

In sum, the $\Rightarrow$ (or $\vDash$) is what you should use when thinking about logical implication, rather than the $\to$.