For residential dwellings
First we do not use a 2 phase system, that proved ineffective years ago or maybe it was politics.... What we use in residential is a transformer with a secondary of 240 volts single phase centertapped step down type. Primary voltage is generally around 2.4 KV to 7.2 KV from a 3 phase source. This allows us to serve both 120 volt loads and 240 volt loads. So our systems for single family dwellings are actually 120/240 volt single phase 3 wire sometimes called the edison system.
This is a common pole mount with diagram graphic yours vs. ours
What actually happens is we split off 2 single phase voltage legs at the top and bottom of the secondary winding for +120 volts and -120 volts with a center tap so total potential between the top and bottom with respect to each other is 240 volts. V1 and V2 with respect to the neutral is + or - 120 volts the resulting wave form (both legs 180 degrees offset from one another) allows cancellation of current in the neutral given perfectly balanced loads otherwise it carries the difference between legs. If you look at the image there are 3 wires.. left and right are the hots V1 and V2 with respect to the graphic. The center is the midpoint tap (neutral) also notice the casing bond then the center tap is earthed with a grounding wire running down the side of the pole to a ground rod.
In a balanced load situation driving two 120 volt loads between V1 and V2 would be like this...using a double pole breaker or two single poles connected to opposite legs at the panel. Notice also that in this configuration we can drive both 120 and 240 volts loads on the same circuit. You can substitute smaller values for amperages if you like.
and 240 volts not using the neutral
Notice the 180 offset but it is not 2 phase it is a split single phase source. So essentially each leg returns on the other to the transformer as the waveform cycles between polarities.