In this answer, we elaborate on the various definitions of integrability, separability & AA-property, in order to expose their (slight) differences.
Let there be given a finite-dimensional autonomous Hamiltonian system, defined on a connected $2n$-dimensional symplectic manifold $({\cal M},\{\cdot ,\cdot \})$.
Definition. The system is (completely) Liouville integrable if
there exist $n$ functionally independent, Poisson-commuting, globally defined functions $F_1, \ldots, F_n: {\cal M}\to \mathbb{R}$, so that the Hamiltonian $H=H(F)$ is a function of $F_1, \ldots, F_n$, only. See also this related Phys.SE post.
Definition. The system is (completely) $H$-separable if there exists an atlas of Darboux coordinates
$ q^1, \ldots, q^n, p_1, \ldots, p_n : {\cal U}\to \mathbb{R}$ with separation functions $F_1, \ldots, F_n: {\cal U}\to \mathbb{R}$
on triangular form
$$F_1~=~F_1(q^1,p_1), \qquad F_2~=~F_2(q^2,p_2; F_1), \qquad F_3~=~F_3(q^3,p_3; F_1,F_2), \tag{1}$$
$$\qquad \ldots, \qquad F_n~=~F_n(q^n,p_n; F_1,\ldots, F_{n-1}), $$
such that the Hamiltonian $H=H(F)$ is a function of $F_1, \ldots, F_n$, only.
Note that the separation functions $F_1, \ldots, F_n$ from Definition 3 are automatically Poisson-commuting and constants of motion, but not necessarily functionally independent. A globally defined $H$-separating Darboux coordinate system with functionally independent separation functions implies integrability.
Theorem. Integrability $\Rightarrow$ $H$-separability. Proof: Use Caratheodory-Jacobi-Lie theorem to extend the Poisson-commuting coordinates $(F_1, \ldots, F_n)$ into an atlas of Darboux coordinate neighborhoods. The Hamiltonian $H(F)$ is then on separable form. $\Box$
Definition. The system is called (completely) $W$-separable
if there exists an atlas of Darboux coordinates
$ q^1, \ldots, q^n, p_1, \ldots, p_n : {\cal U}\to \mathbb{R}$ and a
Hamilton's characteristic function $W: {\cal U}\times \mathbb{R}^n\to \mathbb{R}$ of the form
$$ W(q;\alpha)~=~ \sum_{k=1}^n W_k(q^k;\alpha_1, \ldots, \alpha_n),\tag{2}$$
where $\alpha=(\alpha_1,\ldots,\alpha_n)$ are $n$ independent integration constants, and where
$$ p_k~:=~\frac{\partial W}{\partial q^k}, \qquad k~\in~\{1, \ldots, n\}. \tag{3}$$
such that the Hamilton-Jacobi (HJ) equation
$$ H\left(q,\frac{\partial W(q;\alpha)}{\partial q}\right)~=~h(\alpha)\tag{4} $$
is satisfied. Here $h:\mathbb{R}^n\to \mathbb{R}$ is a given function.
Case where $W$-separability $\Rightarrow$ $H$-separability: Assume that the $n$ integration constants $\alpha=(\alpha_1,\ldots,\alpha_n)$ can be identified with Poisson-commuting separation functions $F_k(z)$, $k\in\{1, \ldots, n\}$. Then $H=h(F)$ and the separation functions become constants of motion.
$H$-separability does not necessarily imply $W$-separability as there is no guarantee that a globally defined Hamilton's characteristic function $W$ exists as a solution to the HJ equation.
Definition. The system has the AA-property if there exists an atlas of angle-action coordinates $(w^1,\ldots, w^n,J_1,\ldots, J_n)$, where the symplectic $2$-form $\omega=\sum_{k=1}^n\mathrm{d}J_k\wedge \mathrm{d}w^k$ is on Darboux form, where each AA-coordinate system is $w$-complete, and where the Hamiltonian $H=H(J)$ does not depend on the angles $w^k$. (We allow non-compact "angle" variables. The compact angle variables has unit period $w^k\sim w^k+1$.)
The AA-property clearly implies all the separability conditions. A globally defined angle-action coordinate system implies integrability.
Case where integrability $\Rightarrow$ AA-property: Assume $\forall f=(f_1,\ldots, f_n)\in\mathbb{R}^n$ that the level sets $\bigcap_{k=1}^n F_k^{-1}(\{f_k\})$ are compact in ${\cal M}$. Then the Liouville-Arnold theorem shows the AA-property. For a proof of the Liouville-Arnold theorem, see my Phys.SE answer here.
