If the current goes the other way, one side sees the impedance of the amplifier. The other (ground) sees the the impedance of the amplifier, but also sees a parasitic path through through that ground and out to "somewhere". Now you have a differential voltage caused by the common mode noise.
What do you mean current goes the other way? It is the same current - not two different cases. If it cancels at the speaker (no current thru speaker) it means net current in the loop is zero. It will also be zero on the amplifier side (no current in the wires). Impedance in the loop is the same for both wires - you cannot separate them - wires are in series. Presence of the speaker doesn't change anything - we can short it. It is the same wire where two induced currents flow in the opposite direction. What might flow to ground is differential current that is zero.
Draw two exactly same current sources of opposite polarities in series and close the loop with resistor. Current in the loop, as well as voltage across resistor will be zero. You can ground it at any point and it won't make a difference - it is floating circuit (no reference to anything). Only connecting it at two points will unbalance currents (alternative path).
As for the case of phono cartridge - as long as it is grounded on one side only it is still loop. When you ground one end of the cartridge it won't make a difference as long as the other end is floating, for instance transformer. It is the same case as with our headphones example - cartridge is inherently balanced and twisting wires will make sum of induced currents zero. If it is zero at the cartridge it will be zero at the amp - no matter if and where you ground one of the wires. Advantage of balanced input will be rejection of common mode noise picked by the cartridge. Perfectly twisted wires will offer great common mode rejection in either case - balanced or single ended.
There are other reasons for driving headphones with balanced drive. One is to get double voltage (often needed with 600ohm headphones), another is to avoid crosstalk between channels by using common return (GND).
@jumia Sorry, for hijacking your thread - I won't write anymore. To your question I can only say what I think is right. I would use two wires with shield connected on the source side only. Many SE cables have an arrow pointing direction from source to receiving end. You can also unscrew shells to see and mark the end with shield soldered to case (GND).