Those are your definitions. All you've done is read some forum posts and, in fact, the forum posts disagree with you, at least many if not most of them.
You call it what you want. I'll call it two phases because it is two phases at the panel, irrespective of the fact that only a single phase was generated. The NEC and those in favor or calling it single phase focus on the generation end, largely to differentiate it from an older two phase system, I'll continue my focus on the user end where, as I stated in my very first post on this, there are two phases seen. The only way around calling it two phase at the user end is to define inverted waveforms as not being phase shifted. I don't agree with that. I don't agree with changing well-established definitions on phase relationships.
I'm not talking about motors. The panel has two phases, phase A to ground and it's mirror opposite, phase B to ground. The panel is one big MWBC and the neutral wouldn't carry the unbalanced load current unless the two hot legs were of opposite phase. The load current would be additive, rather than destructive (subtractive) if A and B were on the same phase.
They aren't my definitions, they are the definitions. As you pointed out, it is the way the NEC defines them. I am not the NEC.
Your definitions are your definitions. Mislabeling a leg as a phase is apt to cause confusion. If a single phase system is two phase in your definition, what do you call an actual two phase (mostly obsolete) system?
Those posts showed the confusion others were having until those who knew explained it to them.
A motor is an actual machine. The machine is ruled by the actual physics. It is not subject to the whim of where the scope leads are placed.
An electric motor in a single phase 120 or 240 volt circuit will not start unless it has additional circuits because there is no actual phase lag that creates the revolving magnetic field to start it.
This is inescapable. If there were real phases available, then these starting circuits would not be necessary as they are not necessary when real phases are available to start the motor.
Real two phase power, motor starts.
Real three phase power, motor starts.
Single phase power, motor needs a start capacitor, or some other clever trick to start it. The Leeson link does a good job of listing those tricks.
It doesn't matter whether you were talking about motors or not. The motor is an independent judge that is swayed only by physics and not frame of reference. The motor says that single phase mid-point neutral is still single phase.