No, it is not two phase power.
Yes, actually it is. Any system with 3 or more wires in which two or more (called phases) are at potentials with respect to one designated as the 'neutral' which peak at different times is a polyphase system. If there were a total of 4 wires and their voltages with respect to the neutral peaked 1/3 of a cycle apart I think you'd agree that we have a 3 phase system. Now suppose there are 2 wires and the voltage on one phase wire peaks 1/4 cycle later than the voltage on the other. This is a biphase system with 90 ° shift. Power used to be generated and distributed this way. Each phase was developed by a coil on the stator of a generator and these were at right angles. If I took a notional generator in which the coils could be moved to any position and shifted the one at right angles to 120 ° we'd have biphase with a 120 ° phase relationship. Thus is exactly what is done in 3-phase generation except that an third coil at 240 ° is also added and that is connected to the 3rd phase wire (we're ignoring number of poles considerations here). If I moved the 2nd coil to 179 ° we'd have a biphase system with a 179 ° relationship and if we moved it to 180 ° then we'd have a biphase system with 180 ° phase shift.
As my generator is a bit impractical and I'd have to build it I'd probably just buy a three phase generator, connect one end of one stator coil to neutral, connect the other end to a red wire, then take the coils from the other two phases and connect them in series with one end of the series chain connected to neutral and the other to a black wire. This would give me biphase with a 180° phase relationship. I actually worked briefly at facility where someone installed a trailer wired for 120 biphase 60 Hz and the facility had 408 three phase 50 Hz. They got the 120V biphase they needed by wiring the generator on an MG set exactly as I described.
Simpler than generators or MG sets is to center tap the coil of a secondary of a transformer. You still get a neutral and 2 phases and its still biphase with a 180° relationship. The fact that you can get it by center tapping a single winding doesn't somehow remove the biphase properties.
It is a single, split phase.
You can get it by 'splitting' a phase but you can also get it by combining 3 phases or by combining 2 as with my hypothetical generator or by wiring two inverters to a common (neutral). In your house you may know that it came from splitting a phase but at my overseas site it didn't and if you are in an industrial setting or large apartment building it probably comes from 2 out of 3 phases of a Y connected 3 phase distribution with the neutral being the neutral of the 3 phase system (in which case you would have two 120 V circuits and/or one 208 V circuit).
Just because lots of people call the legs "phases" doesn't make the correct.
What makes it correct is wide acceptance by engineers who work with this stuff. And it's probably in the IEEE dictionary because I know I didn't invent it.
Almost everyone refers to the neutral wire in a 120V circuit too, but technically it's the grounded wire. There is no neutral wire in a 120V circuit yet that's the parlance.
The NEC calls the white wire the 'grounded conductor' and the bare or green wire the 'grounding conductor'. Little wonder that most of us prefer to refer to the neutral as the neutral.
L1 and L2 being 180 out of phase is merely a matter of the reference point. Using the full winding of the transformer to obtain 240V is decidedly single phase with no relevance to 180 in or out of phase.
Yes that's so and once that reference is available you have created a polyphase system as I explained above at some length above.
The system is 240V, center tapped, which facilitates also providing two 120V legs;
As I noted above I have worked in several places (including here in the US) where it is
not 240V center tapped. To assume that it always is (though that may almost universally be the case for home brewers in the US - I have been to one country where I noticed that the homes are wired 3-phase) is not only incorrect but obscures the fact that a split phase system is indeed a polyphase (biphase) system even in this case.
...not two 120V legs that can also provide 240V.
But that's exactly what they are that's exactly how they are used in house wiring! There are two windings on the transformer on the pole that are connected in series to provide 240 and are used separately to provide 120 circuits relative to the point where those windings interconnect. The two windings may be made by spooling on half the turns, bringing out a loop and then winding the other half of the turns but they are in fact two series connected windings with their common point available.
Perhaps it's all just semantics. If I point out a dog to you and you say no, it's not a dog, it's a Leonberger you are right to some extent but the fact of it being a Leonberger does not mean it is not a dog.