(EDIT: Beware, my definition of torsion-free is incorrect below. The correct one is to say $\mathcal{F}$ is torsion-free if for all $x \in X$, and all $s \in \mathcal{O}_{X,x}-\{0\}$, the multiplication by $s$ map $\mathcal{F}_{x} \to \mathcal{F}_{x}$ is injective. This resolves the issue.)
Let $X$ be an integral Noetherian scheme, and let $\mathcal{F}$ be a coherent sheaf on $X$. One can consider the morphism of $\mathcal{O}_{X}$-modules
$$\mathcal{O}_{X} \to \mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{F}, \mathcal{F})$$
mapping a function $f \in \mathcal{O}_{X}(U)$ to the multiplication by $f$ map $\mathcal{F}|_{U} \to \mathcal{F}|_{U}$. The kernel is called the annihilator ideal sheaf, and the corresponding closed subscheme is $\text{Supp}(\mathcal{F})$.
I want to define $\mathcal{F}$ to be torsion-free when the annihilator ideal sheaf vanishes, and torsion otherwise. I'm able to show that:
If $\mathcal{F}$ is torsion-free, then $\text{Supp}(\mathcal{F})=X$.
$\mathcal{F}$ is torsion-free if and only if $\mathcal{F}_{x}$ is a torsion-free $\mathcal{O}_{X,x}$-module for all $x \in X$.
$\mathcal{F}$ is torsion-free if and only if it has no non-trivial torsion subsheaves.
Now, the converse of (1) is obviously false: if $\mathcal{F} = \mathcal{L} \oplus \mathcal{O}_{p}$ is the direct sum of a line bundle and skyscraper sheaf, then $\text{Supp}(\mathcal{F})=X$, but $\mathcal{F}$ is not torsion-free. However, since the annihilator ideal sheaf should be the ideal sheaf corresponding to X, it must be zero? Contradicting that $\mathcal{F}$ isn't torsion-free? Moreover, in this example above, it seems like the annihilator ideal sheaf should vanish at all stalks except $p \in X$. Which cannot happen for an ideal sheaf. Where am I going so badly wrong here? Is it related to support vs. scheme-theoretic support?
My second confusion is with showing that my definition of torsion-free is equivalent to the rank $rk(\mathcal{F}) = dim_{\mathcal{O}_{X, \eta}}\mathcal{F}_{\eta}$ being larger than zero, where $\eta \in X$ is the generic point. I think its clear that my definition implies this one. But conversely, how can one show that $\mathcal{F}_{x}$ is torsion-free for all $x \in X$ just from this data at $\eta \in X$? I know that all open sets of $X$ contain $\eta$; it must be using that somehow?