And all this time, there is a single current flowing through a single transformer winding.
Even if you had not removed one of the loads, why would you think both halves of the winding would have the same current?
While the example loads you used were single-phase, consider that a two-phase load could have current phasors that are the same as two unbalanced single-phase loads. Are you saying the X23-X1 current would be the same as the X23-X4 current?
So, sources and supplies are not the same thing.
They are not the same. My source may be a three-phase transformer bank. I may use it to supply single-phase to a load. That does not mean I no longer have a 3-phase source.
I may take out three separate single-phase systems from separate source phases. I am using it as a source of single-phase supplies. That does not mean I no longer have a 3-phase source back at the transformer.
What it means is that I have three separate single-phase sources brought from the transformer. If I have split the single-phase sources out to separate properties, then at the cabinet on the properties I would say I have a single-phase source at the cabinet. If these single-phase sources traveled together and were in a single cabinet on a common property where they could be re-combined, then I would say I have a three-phase source at the cabinet (i.e., three single-phase sources that can combine to make a three-phase source). While you could say you have single and three phase, I have indicated the highest order available which is what we normally do.
You have changed the restricted sets of supplies that you can take from the source. By stipulating that X1 must be the reference, you are defining the system that the source will supply.
So, again the choice of the reference defines if you have two 1-phase or one 2-phase.
It tells me if 2-phase is one of the available choices and determines the highest order available. Without it, the highest order available is single-phase. With it, the highest order available is two=phase.
Aren't you saying, that the number of phases depends on if the neutral is used or not?
The number of phases that can be taken out is dependent on whether or not the center-tap neutral is used
Clearly, this says the way loads are wired defines the number of phases.
It certainly defines what is being used. It does not define what is available at the source
The only thing that changes is the use of an arbitrary reference point, but you don't consider that 'a sleight of hand'.
It is what goes into defining the system. Defining a system is not 'sleight of hand'. We must define a reference frame from which to take measurements. If the load requires both a +V and -V source and the neutral point between them, there is nothing arbitrary about it. If I have some independent loads at V and some at 2V, any reference will do.
I guess it comes down to:
You are defining 2-phases based on how things are wired/used.
I am defining 2-phases based on how the current is being generated/created based on conductors in magnetic fields.
I am defining a 2-phase source based on what kind of voltages are available in the system. A normal n-phase system will have n voltages evenly displaced around 360 degrees. An abnormal n-phase system will have n voltages that are displaced by some angle, but the voltages are not evenly displaced around 360 degrees.
Do you think the currents in a split winding would be any different than the currents in two separate windings using separate but synchronized fields?