Reduced mass effect on gyromagnetic factor

[Khashishi]

Is there an effect of reduced mass on the orbital g factor? In Bethe and Salpeter (1957) Quantum Mechanics of One and Two Electron Atoms p214, it mentions a reduced mass effect on $g_L = \frac{\mu}{m_e}$. (At least, that's what my notes said. I don't have Bethe and Salpeter in front of me now.)
But I can't find mention of it anywhere else. Wikipedia says that $g_L = 1$ exactly.

So I tried to sort it out by using positronium as an example. Now, for the case of positronium, the energy levels are about half of hydrogen, so the average distance between the electron and positron is twice the distance between the electron and proton. Basically, they orbit the center of mass at the same distance. The 2p state has $l=1$ and $L=\sqrt{2}\hbar$. The orbital angular momentum is equally shared between the positron and electron, so the average orbital velocity of each is about half of the orbital velocity of the electron in the hydrogen atom. This means the current from the electron is half, and the magnetic field should be half also. Right? But also, the positron has a current which cancels the electron current, so the orbital g factor should be 0? Did I think that through correctly?

 

[Khashishi]

Lamb. Physical Review 85 2 (1952). eqn 153 shows an effective $g_L = 1-(1/M)$ where I guess M is the mass ratio. So it agrees with Bethe and Salpeter. But I'm still bothered by the positronium example.