Please let me know if I am wrong, but isn't it true that when you measure voltage of any kind, you are actually measuring it with respect to something- either to ground, or to a lesser or greater voltage? In other words, you are measuring a difference in voltage. Isn't that correct?So when you measure one leg of a 240vac receptacle for example, you are measuring the difference in voltage between the top of the wave and the neutral or midpoint?
If I understand your question correctly,, Yes & No.. when you measure 120Vac, you are measuring the wave to to neutral or zero line.
240Vac is measure total swing (peak to peak) and neutral or zero line is not involved.
Does your 240 volt line have four wires or three? You need four to get 120v without improperly using the ground as a neutral.
None of my comments have involved ground, fault return or green wire,, yet..
I hope the O P went to a new breaker on the panel for his 120Vac. In re-reading the O P post #1, I am now concerned.
If the O P pulled 120Vac off of the clothes dryer outlet, this would be improper and a concern for a few reasons. There is no Neutral lead in the old Crows Foot 3 wire out let. Using the green lead for neutral is against every electrical code there is before ABYC was ever created. This is an example of why the new dockside GFI breakers are popping.
If it was a 4 wire lead with the white and green wire separate, and the white (neutral) and red or black lead used for 120Vac, it would work,,
PENDING; This modification was
built as a sub panel with a new dryer breaker next to the washer breaker, The breaker at the main panel upgraded to supply the new total current requirements
and the wires between the main panel to the new sub panel are of large enough in size to carry to spec, the additional current required. If none of these are so, it is not safe and not to any code.