Now that's a lathe! Hot damn, Erik!
That neighbor is lucky he doesn't live next door to me. He'd be walking funny and sleeping face down for awhile, and that's after the doctors strap him to the bed and remove the camera from up his ass with crane and a pair of ice tongs. Nothing brings out that war vet rage in me like someone thinking they can threaten my peaceful pursuit of my own goddamned ambitions. I even scare the cops, too, making the whole scene a little confusing. You shouldn't follow suit, that's just me saying that type of nosiness is over the line, lest your neighbor need to be corrected.
Noise control shouldn't be much of a thing with a lathe. Insulation in the walls goes a long way to adding sound damping mass. I got over 20 dB of attenuation by installing 6" of fiberglass in my shop. It looks like your door is insulated too. They use either glass fiber or rockwool in those doors, rockwool (green color) being the better sound insulator. If you're really worried about it, you can face the inside of the garage door panels with 1" styrofoam panels, which will help more with high frequency noise than it will with low freqs. That's kind of a tacky solution for the door, since you already have a good one.
Another noise control option is to play great music (Slayer, Cannibal Corpse, Sepultura, whatever) at just under the legal limit while working in your garage. Suddenly the low whummmm of a lathe might not be such a big deal. And when you do see your neighbor, snarl at him. Make him tuck tail.