If you connect a winding as an autotransformer between the two 120 to volt ground legs then the midpoint of that winding will be near ground potential, only because midpoint of the source end is grounded.
If you connect same winding between one 120 volt to ground lead and the high leg - you still have 120 volts to the mid point, from each ungrounded conductor, but one of them is 208 volts to ground at the supply end. Not fully sure but just doing a little drawing of some triangles to represent the supply and where that tap would end up at tells me it would be 120 volts to ground from our so called "derived neutral".
If A and C are the two ends of the source center tapped 240, then if your autotransformer runs from A to C you will have the midpoint, M, near the grounded neutral of the source. True.
If you run your autotransformer from either A or C to B, you will indeed have 120V from M to source neutral. An interesting result all right.
But that is not particularly relevant since we have presupposed that the machine derives its 120V either from A to neutral or from C to neutral.
If you try to derive your neutral from other than AC, then you will have to make sure that the machine also expects the neutral to be the midpoint of the same leg that you connected to.
And in that case for sure you should not ground your derived neutral!!!!