Not who you asked, but after looking into it it’s because the moon takes about 29.5 days to complete a full cycle of phases (one synodic month), giving it time to do so roughly 12 times per year.

I can’t quite wrap my head around it, but I think the explanation for why sidereal and synodic months differ lines up with your initial explanation. Because we’re also moving, the moon has to move further to achieve the visual change of moon phases.