I will try to link the two concepts in the following way. It is rough, descriptive, but contains the essential idea.
Here we are talking about the so-called orbital angular momentum (OAM) of an electron orbiting around the nuclei. In classical mechanics, OAM is defined as
$
\vec{L} \equiv \vec{r}\times\vec{p}
$
where $\vec{r}$ is the position of the electron, and $\vec{p}$ is the momentum. One could roughly think that the magnetic moment to be proportional to $\vec{L}$, namely $\vec{M}\propto\vec{L}$ given that we ignore the spin part (which are in fact equally important). Now in quantum mechanics, the OAM is given by pretty much the similar expression, except two things. First, $\vec{r}$ and $\vec{p}$ should be thought as operators now (see below) and second, the operators are sandwiched between wavefunctions. Thus
$
\vec{L} = \langle \psi | \vec{r}\times\vec{p} |\psi \rangle
$
Precisely what this means is that, if you want to calculate say the component $L_z$, then the expression is
$
L_z = i\int d^3r \psi^*(r) ( x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x} )\psi(r)
$
where we have written out the operator $\vec{p}$ explicitly as $i\frac{\partial}{\partial \vec{r}}$ for the purpose of calculation.
That said, the two point of view, one with orbiting electron and the other with probability cloud is not in conflict with each other. Rather, they are complementary in the sense that the former view is more intuitive for us to imagine the physics and write down equations, but when it comes down to actual calculations, we have to use the wavefunctions.
