I have no idea where the so called Return wire or conductor name for the Neutral came from but that is completely misleading.
Most likely it came from using sign conventions where one lead is designated as the one where current is leaving and the other as the one where current is returning. While the choice is arbitrary, we pick a point in time and by normal convention use a positive terminal with an outgoing positive current. Since the neutral is grounded in most cases, the natural progression of convention would be to call the neutral the common return for both circuits.
That is no different than in a four-phase system we note that the four emfs of the quadrature system have opposing pairs and thus can be produced with two coils. The choice of which phase is the return of the other is arbitrary. The non-symmetrical two-phase system with a 90 degree displacement is really just a sub-set of the symmetrical four-phase system.
One might note that the four-phase system, can be produced from a two-phase supply using center-tapped windings. That is, we produce four emfs from two emfs. These are four emfs of the four-phase system even though we can group them into two opposing pairs.
Another one is calling single phase lines Phase A and Phase B instead of lines #1, #2 and #3 if three phase
For a two-bushing transformer, they very well may be the same as the primary phases A and B. The crux of the matter is the definition of phase.
The fundamental definition is that any point on an alternating waveform is a phase. We loosely use this term to apply to an alternating emf or current in a circuit. With a single phase emf or current there is only one phase at any instant in time. Using opposing waveforms, we can have two waveforms with different phases present at any instant in time.
This becomes more obvious when one considers that with a 2-wire circuit, there is only current present: the outgoing on one wire is the incoming on the other wire so we only have one phase. For a 3-wire circuit, the current leaving line #1 does not have to equal the current entering line #2. In fact, the currents may have different magnitudes and may also flow at completely different times. The obvious case would be loads only between line #1 and neutral. Another is for a push-pull circuit. Not so obvious is when we have mixed or balanced loads.
these terms became popular only after the Internet became popular and the folks in the Electronic field and Electrical field got closely involved together.
When you study Electronic the split phase transformers to them is a phase converter, and it never even really did that it just give them a inverted voltage to use because the so called phase inverter needed a different polarity to work with. These folks started telling all the Electrical folks
well that's just a phase inverter so its got to be two Phase.
Not true, the terms have been around since the beginning:
These uses have been around as long as multiple phases have been around. I have references dating back to the turn of the century and, if I remember, back to the mid-to-late 1800's.
These include C.P. Steinmetz who said we can take the negative phase to be the return circuit of the positive phase, producing a larger single phase circuit. While discussing the symmetrical four-phase system he noted that it could be produced by two coils in quadrature with each other (as noted above). He also called the ordinary alternating current system a two-phase system that could be produced with one coil.
It also includes Terrell Croft who wrote the American Electrician's Handbook back in the early 1900's. Other old texts also do the same covering AC theory, AC machines, transformers, power system analysis & engineering, antennas, and audio as well as electronics.
Some of the original realization came about around the 1860's when experiments were conducted to see if they could combine two smaller forces to create a larger force. They found they could combine both in-phase forces and phase-opposed forces. Either way, the result was one larger single-phase force. We take it for granted today that two in-phase voltages can be combined to produce a larger single-phase voltage. We also know two phase-opposed voltages can be combined to produce a larger single-phase voltage.
A limited view leads to limited understanding:
The limited view is that there is only a single phase. The reality is that in the more general case the center-tapped transformer can serve as both a source for a single emf or for phase-opposed emfs.
This fact is used for supplying two-diode full-wave rectifiers. The fact that we can draw current at different times for each half of the winding is evidence that the currents are not just mirrors of each other and can be different in phase. Consider that a normal single-phase circuit the instantaneous power delivered to the load pulses at twice the line frequency. Now consider equal pulse circuits on two windings. If both pulse at the same time, combining the two produces a larger pulse at line frequency. If they were pulsing on opposing halves of the cycle, combining the two produces equal pulses at twice the line frequency.
If one looks at the double frequency result alone they might lose sight of the fact that there are two separate circuits and that these two circuits could differ in phase even if a particular case makes them look like something else.
Anyway, the fact that the center-tapped transformer can be a source for phase-opposed emfs is also used in creating high phase order systems and other neat things like creating a 4-wire 3-phase wye from an open-wye transformer.
In the general case, the center-tapped transformer is a source for series additive single-phase forces as well as phase-differentiated forces. Both sources map to the same physical space and to ignore one is to limit one's scope of understanding.
But a limited understanding is just fine for most:
A limited understanding is fine and will serve most people all their life. We see similar instances in physics where most remember the limited equation E=MC^2 but forget that it is not true for the general case.
As engineering students we were driven to remember that force equals mass times acceleration (F=ma). That works for most of the stuff we need (low velocities) but we may forget that it is not true for the general case (where we need force equals the rate of change of momentum with respect to time or F=dP/dt).
