How can a 240V single-phase split ac work in the US which has a 120V 2-phase system?
"2-phase" to describe US residential service is depricated nomenclature. US residential service is single phase because the two hot supply phases have 0 degrees of separation between their phases: therefore they are in phase.
The two phases are pulled from 2 out of 3 corners on a 240v (line-to-line) 3-phase delta-primary transformer, usually out in the street. (Sometimes at your house if it's a big one). Basically all power is generated and transmitted as 3-phase. Usually around 415V 3-phase comes into the primary of the 3-phase transformer. (It could be delta- or wye-connected.) The secondary windings for all three phases will each have a center-tapped connection, and that's grounded at the panel. This is the neutral wire. All circuits in houses return via the "common" or "neutral" wire to this point, which happens to be grounded at the panel. Note: no neutral wires should be grounded anywhere else. The two "corners" are 240v relative to each other, and 120v each to the center tap. Each 240V circuit really is two hot wires plus a neutral, which is why it's often wrongly called two-phase when in fact it's just single-phase. For each 120V circuit in the house you're going to use one hot wire and one neutral wire. The neutral wire provides a return path for currents back to the generating station. In some installations, such as apartment blocks, the third leg of power would be at 208V with respect to the center tap.
Just what does "220v single-phase split ac in the US which has a 110v 2-phase system" mean?
The vast majority of electricity in the US is delivered as single phase or three phase. The only areas in the US that use 2-phase [for industrial and commercial purposes] are Philadelphia/South Jersey [where it is being phased out-no joke intended] and somewhere out west...
It was one of the early poly-phase options pursued because of the natural magnetic differential between phases [makes motors spin without a capacitor]
Most residences receive 220-240 volt single phase electricity with a grounded center-tapped neutral, the purpose being to limit voltage to ground to less than 150 volts from either "hot".