Gravity has no bearing on either.
For example if you produce enough anti-gravity to counter act all the gravity your ship produces then you still have all of the mass and inertia that goes with it.
For example: Just because you are in zero gravity, that doesn't mean you can pick up the space shuttle with your hands, it will remain fairly motionless. Or, if the shuttle and the ISS were about to smash into each other, you couldn't stand between them (because they both have effectively zero gravity) and push them apart, you'd be crushed because of their inertia imparted by their mass.
Inertialess drives on the wiki don't even use anti-gravity to generate this effect, they in essence cancel out the mass of the ship with particles that have negative mass (Tachyons). Of course this wouldn't work in real life either, but that doesn't matter because this device does not work in this way.
By generating an anti-gravity field (which is what this device does) the only result it would have would be to push everything away from the ship. It would not be able to accelerate the ship without help from the thrusters nor would it help reduce the effects of acceleration on the crew because it can only uniformly push them all away from itself, not distribute a gradient of force to help reduce the shock of being accelerated rapidly.
Bottom line: You are both confusing Weight with Mass.