Because single-phase is bipolar, it defines two rotating vectors 180deg apart: motors demonstrate that perfectly, single-phase motors have two set of coils driven 180 deg apart, which push and pull alternatively. There is no concept of split-phase motor because it adds exactly nothing: the same "two phases" 180deg apart can be had by simple polarity reversal from a single supply!
Two-phase motors exist, most notably capacitor-run single-phase motors, stepper motors, and some BLDC fan motors. Distinctive feature is the 90deg phase shift: the bipolarity of the signal itself creates the two more 180deg out-of-phase versions; so a 1-phase motor has 2 poles, 2-phase motor has 4 poles. And yes - in some cases, some people call 2-phase motors 4-phase motors, the same confusion.
That's the practical version of the theory.
Note the difference to DC/DC buck converters: 2-phase buck does have 180, not 90 deg phase shift. This is because the construction is unipolar. In DC/DC buck, 180 deg phase shift doubles the ripple frequency. In bipolar, sinusoidal AC systems - think about full-bridge rectifier - 180 deg out-of-phase separate supply adds exactly nothing (in full bridge diode example, diodes already utilize both polarities).
Split phase AC is like two batteries in series. It offers a center tap for a half-voltage point, nothing more, that's it. Putting 3 AC supplies in series does not make 3-phase power, either.
Transformers with a lot of taps are plentiful. No one in their right mind would call them 5-phase or 9-phase power supplies.
Tapping out intermediate voltages out of AC supply does not generate "phases". The mental trick of choosing one of them as ground/reference level does not change this.
There are people, especially electricians, who use the word "phase" more loosely, kind-of replacement for the word "circuit". They are free to do so. With a split phase (3 wires) entering the house, saying "phase A" and "phase B" is unambiguous, and unnecessary pedantism is tiresome. But here, the OP wants to understand what it really is, and for that, the answer is simple, split-phase is regular single-phase AC with a center tap for half the voltage. That's it.
But the word split-phase exists exactly because it isn't 2-phase system. The word "split" itself suggests that the origin is singular: a single one is being split. 2-phase power distribution did exist. The words are not random, they carry real meanings and history.